Event-driven spatio-temporal short-time fourier transform processing for asynchronous pulse-modulated sampled signals

ABSTRACT

A method of processing asynchronous event-driven input samples of a continuous time signal includes calculating a short-time Fourier transform (STFT) output based on the event-driven input samples. The STFT output may be calculated by expressing an encoding pulse and an STFT window function as a sum of complex weighted causal complex exponentials. The method further includes interpolating output between events.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims the benefit of U.S. Provisional Patent Application No. 62/045,775, filed on Sep. 4, 2014, and titled “EVENT-DRIVEN SPATIO-TEMPORAL SHORT-TIME FOURIER TRANSFORM PROCESSING FOR ASYNCHRONOUS PULSE-MODULATED SAMPLED SIGNALS,” the disclosure of which is expressly incorporated by reference herein in its entirety.

BACKGROUND

Field

Certain aspects of the present disclosure generally relate to machine learning and, more particularly, to improving systems and methods of event-driven spatio-temporal short-time Fourier transform processing for asynchronous pulse-modulated sampled signals in neural networks.

Background

An artificial neural network, which may comprise an interconnected group of artificial neurons (e.g., neuron models), is a computational device or represents a method to be performed by a computational device.

Convolutional neural networks are a type of feed-forward artificial neural network. Convolutional neural networks may include collections of neurons that each have a receptive field and that collectively tile an input space. Convolutional neural networks (CNNs) have numerous applications. In particular, CNNs have broadly been used in the area of pattern recognition and classification.

Deep learning architectures, such as deep belief networks and deep convolutional networks, are layered neural networks architectures in which the output of a first layer of neurons becomes an input to a second layer of neurons, the output of a second layer of neurons becomes and input to a third layer of neurons, and so on. Deep neural networks may be trained to recognize a hierarchy of features and so they have increasingly been used in object recognition applications. Like convolutional neural networks, computation in these deep learning architectures may be distributed over a population of processing nodes, which may be configured in one or more computational chains. These multi-layered architectures may be trained one layer at a time and may be fine-tuned using back propagation.

Other models are also available for object recognition. For example, support vector machines (SVMs) are learning tools that can be applied for classification. Support vector machines include a separating hyperplane (e.g., decision boundary) that categorizes data. The hyperplane is defined by supervised learning. A desired hyperplane increases the margin of the training data. In other words, the hyperplane should have the greatest minimum distance to the training examples.

Although these solutions achieve excellent results on a number of classification benchmarks, their computational complexity can be prohibitively high. Additionally, training of the models may be challenging.

SUMMARY

In an aspect of the present disclosure, a method of processing asynchronous event-driven input samples of a continuous time signal is presented. The method includes calculating a short-time Fourier transform (STFT) output based on the event-driven input samples. The STFT output may be calculated by expressing an encoding pulse and an STFT window function as a sum of complex weighted causal complex exponentials. The method may further include interpolating output between events.

In another aspect of the present disclosure, an apparatus for processing asynchronous event-driven input samples of a continuous time signal is presented. The apparatus includes means for calculating a short-time Fourier transform (STFT) output based on the event-driven input samples. The STFT output may be calculated by expressing an encoding pulse and an STFT window function as a sum of complex weighted causal complex exponentials. The apparatus further includes means for interpolating output between events.

In yet another aspect of the present disclosure, the image capturing device is presented. The image capturing device includes a memory unit and at least one processor coupled to the memory unit. The one or more processors are configured to calculate a short-time Fourier transform (STFT) output based on event driven samples.

In still another aspect of the present disclosure, a non-transitory computer readable medium is disclosed. The non-transitory computer readable medium has program code thereon, which when executed by processor causes the processor to process asynchronous event-driven input samples of a continuous time signal. The program code includes program code to calculate a short-time Fourier transform (STFT) output based on event driven samples.

Additional features and advantages of the disclosure will be described below. It should be appreciated by those skilled in the art that this disclosure may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present disclosure. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the teachings of the disclosure as set forth in the appended claims. The novel features, which are believed to be characteristic of the disclosure, both as to its organization and method of operation, together with further objects and advantages, will be better understood from the following description when considered in connection with the accompanying figures. It is to be expressly understood, however, that each of the figures is provided for the purpose of illustration and description only and is not intended as a definition of the limits of the present disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The features, nature, and advantages of the present disclosure will become more apparent from the detailed description set forth below when taken in conjunction with the drawings in which like reference characters identify correspondingly throughout.

FIG. 1 illustrates an example implementation of designing a neural network using a system-on-a-chip, including a general-purpose processor in accordance with certain aspects of the present disclosure.

FIG. 2 illustrates an example implementation of a system in accordance with aspects of the present disclosure.

FIG. 3A is a diagram illustrating a neural network in accordance with aspects of the present disclosure.

FIG. 3B is a block diagram illustrating an exemplary deep convolutional network (DCN) in accordance with aspects of the present disclosure.

FIG. 4 is a block diagram illustrating an exemplary software architecture that may modularize artificial intelligence (AI) functions in accordance with aspects of the present disclosure.

FIG. 5 is a block diagram illustrating the run-time operation of an AI application on a smartphone in accordance with aspects of the present disclosure.

FIG. 6 illustrates a method for processing asynchronous event driven samples of a continuous time signal according to aspects of the present disclosure.

FIG. 7 is a block diagram illustrating a method for event-driven short-time Fourier transform processing according to aspects of the present disclosure.

DETAILED DESCRIPTION

The detailed description set forth below, in connection with the appended drawings, is intended as a description of various configurations and is not intended to represent the only configurations in which the concepts described herein may be practiced. The detailed description includes specific details for the purpose of providing a thorough understanding of the various concepts. However, it will be apparent to those skilled in the art that these concepts may be practiced without these specific details. In some instances, well-known structures and components are shown in block diagram form in order to avoid obscuring such concepts.

Based on the teachings, one skilled in the art should appreciate that the scope of the disclosure is intended to cover any aspect of the disclosure, whether implemented independently of or combined with any other aspect of the disclosure. For example, an apparatus may be implemented or a method may be practiced using any number of the aspects set forth. In addition, the scope of the disclosure is intended to cover such an apparatus or method practiced using other structure, functionality, or structure and functionality in addition to or other than the various aspects of the disclosure set forth. It should be understood that any aspect of the disclosure disclosed may be embodied by one or more elements of a claim.

The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any aspect described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects.

Although particular aspects are described herein, many variations and permutations of these aspects fall within the scope of the disclosure. Although some benefits and advantages of the preferred aspects are mentioned, the scope of the disclosure is not intended to be limited to particular benefits, uses or objectives. Rather, aspects of the disclosure are intended to be broadly applicable to different technologies, system configurations, networks and protocols, some of which are illustrated by way of example in the figures and in the following description of the preferred aspects. The detailed description and drawings are merely illustrative of the disclosure rather than limiting, the scope of the disclosure being defined by the appended claims and equivalents thereof.

Event-Driven Spatio-Temporal Short-Time Fourier Transform Processing

Uniform sampling is used in conventional data acquisition and signal processing techniques. The sampling frequency may be determined according to a maximum expected spectral frequency. This maximum-frequency sampling, however, may waste power for input signals with relaxed properties (e.g., reduced frequency content, increased periods of silence). However, the use of maximum-frequency sampling is problematic for emerging applications that rely on scarce energy resources.

Event-based sampling is a promising alternative to uniform sampling. In event-based sampling, a sample is only output when something significant (e.g., an event) occurs in the signal. One aspect of the present disclosure involves an asynchronous, event-driven signal processing technique for powering future, energy-efficient neuromorphic systems that rely on scarce energy resources. For example, the described techniques may be useful in event-based sensors, such as the dynamic vision sensor (DVS).

An aspect of the present disclosure is directed to an event-driven processing for computing the spatio-temporal short-time Fourier transform (STFT) of sampled signals (e.g., Lebesgue and/or asynchronous pulse modulated (APM) sampled signals) in a continuous time and discrete space. In particular, one aspect relates to an asynchronous event-driven processing for a short-time Fourier transform of sampled signals in which an encoding (APM) pulse and the window function in the short-time Fourier transform are expressed and/or approximated as a sum of complex weighted causal complex exponentials. Accordingly, event-driven spatio-temporal short-time Fourier transform processing for asynchronous pulse-modulated sampled signals in neural networks may power future neuromorphic systems that are highly energy-efficient.

In accordance with aspects of the present disclosure, a continuous analog signal may be received as an input. An event-based sampling process may be applied to generate a train of pulses. In some aspects, the sampling process may comprise an APM sampling process or the like. The train of pulses may be positive pulses, negative pulses or both (bipolar). Because the input is sampled on the occurrence of an event, the sampling rate is much lower than may be observed for uniform time sampling, for example. As such, the present systems and methods make satisfying the Nyquist sampling rate to recreate an input signal more readily achievable.

The train of pulses may in turn be processed directly applying a short-time Fourier transform to approximate the time domain output. That is, the train of pulses may be used to generate the output signal without converting the pulses back to analog to process and create the output signal.

Asynchronous Event-Based Sampling

Consider an integrable continuous-time real signal x(t): [t_(o),∞)→

from time t₀, where X is the set that contains all such signals x(t).

A polarized event train ξ(t): [t_(o),∞)→

is a response function of an ordered set of polarized events {(t_(k),p_(k))|t_(k)ϵ[t₀, ∞), p_(k)ϵ{−1,1}}(k=1, . . . , K), such that: t _(k) <t′ _(k) if k′, and  (1)

$\begin{matrix} {{{\xi\left( {t;\left\{ \left( {t_{k,}p_{k}} \right) \right\}} \right)} = {\sum\limits_{k}{p_{k}{\delta\left( {t - t_{k}} \right)}}}},} & (2) \end{matrix}$ where δ is the Dirac delta function. By ≡ we denote the set containing all such event trains ξ(t). The direct reconstruction of a polarized event train ξ(t; {(t_(k),

k)}) under kernel function h(t) may be given by its convolution with h(t),

$\begin{matrix} {{y(t)} = {\left\lbrack {\xi*h} \right\rbrack(t)}} & (3) \\ {\mspace{40mu}{= {\int{{h(T)}{\xi\left( {t - \tau} \right)}d\;\tau}}}} & (4) \\ {\mspace{40mu}{= {\sum\limits_{k}{p_{k}{{h\left( {t - t_{k}} \right)}.}}}}} & (5) \end{matrix}$

In some aspects, the kernel function may be causal (e.g., h(t)=0, ∀t≤0). We further denote by ρh:≡→X the direct reconstructor (e.g., y(t)=ρh[ξ](t)).

An asynchronous pulse modulation (APM) sampler

_(h):X→≡ may transform x(t) to a polarized event train ξ(t; (t_(k),p_(k)))=

_(h)[x](t) under kernel function (e.g., pulse) h(t) such that ∀kϵ{1, . . . , N}:

$\begin{matrix} {{{{{x\left( t_{k} \right)} - {y\left( t_{k} \right)} - {x\left( t_{o} \right)}}} = \frac{1}{2}},} & (6) \\ {{{sign}\left( {{x\left( {t_{k} - 0} \right)} - {y\left( {t_{k} - 0} \right)} - {x\left( t_{o} \right)}} \right)} = {p_{k}\mspace{14mu}{and}}} & (7) \\ {{{{{x(t)} - {y(t)} - {x\left( t_{o} \right)}}} < \frac{1}{2}},{\forall{t \notin \left\{ t_{k} \right\}}},} & (8) \end{matrix}$ where 0 is an infinitesimally small positive number and y(t)=[x*h] (t) is the direct reconstruction of ξ(t) under h(t).

A level-crossing sampler (e.g., Lebesgue sampler)

_(Θ):x→≡ is an APM sampler with a heavy-side step function Θ(t) as its kernel function. The heavy-side step function may be defined as: Θ(t)=∫_(−∞) ^(t)δ(t)dt.  (9) Event-Driven Signal Processing

Given continuous signals x(t), the goal of processing can be, in general, defined as the computation of a transform of the continuous time signals x(t). For example, in some aspects, the transform may be given by: y(t,s)=

[x](t,s),  (10) where s represents optional extra independent variables.

In practice, x(t) may be first sampled by a sampler

: ξ(t;{(t _(k) ,q _(k))})=

[x](t),  (11) which may result in a time-ordered set of time-value pairs {(t_(k),q_(k))}. In some aspects, t_(k)s may be chosen such that they have regular, equal intervals, so as to configure the sampler as a Riemann sampler. In another configuration, the q_(k)s may be binary so as to configure the sampler as an APM sampler. Furthermore, the sampler may be configured as a level crossing sampler (e.g., a Lebesgue sampler).

Then, a technique

is applied to the sampled signal ξ(t): {circumflex over (y)}(t,s)=

[ξ](t,s),  (12) to compute an approximated target transform ŷ(t,s)≈y(t,s) defined at times {t_(k)}. Thus, in general, a signal processing paradigm may be completely determined by a target transform, a sampler, and a technique (

,

,

), that operates on sampled signals.

As a special case, if

is the identity transform (e.g.,

=I) where I is the identity function, the signal processing paradigm reduces to a communication paradigm: y(t,s)=I[x](t)=x(t),  (13) of which the goal is to reconstruct signal x(t) from its sampled version ξ(t).

In some aspects, an event-driven technique may be used to transform ξ(t; {(t_(k), q_(k))}) to ŷ(t,s). If

can be expressed in the following recurrence relation: {circumflex over (y)}(t _(k) ,s)=f

((t _(k) ,q _(k));(t _(k-1) ,q _(k-1) ,ŷ(t _(k-1) ,s)), . . . ,(t _(k-n) ,q _(k-n) ,ŷ(t _(k-n) ,s)))  (14) =f

(t _(k) ,t _(k-1) , . . . ,t _(k-n) ;q _(k) ,q _(k-1) , . . . ,q _(k-n) ;ŷ(t _(k-n) ,s),  (15) where n is a finite non-negative integer, the technique may be considered event-driven. That is, the output value k upon arrival of the sample, or synonymously event, (t_(k),q_(k)), may be computed from the current sample, in addition to a finite history.

For example, if f

can be further expressed as sample time intervals rather than absolute time, {circumflex over (y)}(t _(k) ,s)=f

(t _(k-) t _(k-1) , . . . ,t _(k-) t _(k-n) ;q _(k) ,q _(k-1) , . . . ,q _(k-n) ;ŷ(t _(k-1) ,s), . . . ,{circumflex over (y)}(t _(k-n) ,s)),  (16) the event-driven technique may also be called time invariant.

Further, in some aspects f

may be expressed in the following generalized linear form:

$\begin{matrix} {{{\hat{y}\left( {t_{k},s} \right)} = {{\sum\limits_{j = 0}^{n}{{A\left( {t_{k - j},t_{k},s} \right)}{\varphi\left( {t_{k - j},q_{k - j},s} \right)}}} + {\sum\limits_{j = 1}^{n}{{B\left( {{t_{{k - j},}t_{k}},s} \right)}{\hat{y}\left( {t_{k - j},s} \right)}}}}},} & (17) \end{matrix}$ where φ is a vector function of sample value q's, and A and B are, respectively, the feed-forward and feedback kernel functions. The event-driven technique may be characterized as a generalized linear. Furthermore, in some aspects, a time invariant event-driven signal processing technique f

may be expressed in the following generalized linear form:

$\begin{matrix} {{{\hat{y}\left( {t_{k},s} \right)} = {{\sum\limits_{j = 0}^{n}{{A\left( {{t_{k - j} - t_{k}},s} \right)}{\varphi\left( {t_{k - j},q_{k - j},s} \right)}}} + {\sum\limits_{j = 1}^{n}{{B\left( {{t_{k - j} - t_{k}},s} \right)}{\hat{y}\left( {t_{k - j},s} \right)}}}}},} & (18) \end{matrix}$

As such, the event-driven signal processing technique A may be considered linear time invariant (LTI).

In one example, where Riemann sampling of a single signal (e.g., t_(k)s) are spaced by regular intervals, q_(k)=[x(t_(k))] and φ being the identity function of q_(k), a time invariant generalized linear event-driven signal processing technique may comprise an infinite impulse-response (IIR) filter.

Event-Driven Short-Time Fourier Transform

The Short-time Fourier transform (STFT) is used to determine frequency and phase content of local sections of a signal as it changes over time. The STFT of signal x(t) under window function w(t) may be defined as: {tilde over (x)}(τ,ω)=∫x(t)ω(t−τ)e ^(−1ωt) dt.  (19) STFT of the Direct Reconstruction of an APM Sampled Signal

-   When x(t) is first APM sampled by ρ_(h), e.g., ξ(t)=ρ_(h)[x](t), and     then direct reconstructed by ρ_(h)(e.g.,     y(t)=ρ_(h)[ξ](t)=(ρ_(h)ορ_(h))[x](t)) the STFT of y(t) may provide     an approximation of {tilde over (x)}(τ,ω):     {tilde over (y)}(τ,ω)=∫y(t)ω(t−τ)e ^(−1ωt) dt.  (20)

Using {(t_(k),p_(k))} to denote the event train and applying Eq. 5,

$\begin{matrix} {\begin{matrix} {{\overset{\sim}{y}\left( {\tau,w} \right)} = {\int{\sum\limits_{k}{p_{k}{h\left( {t - t_{k}} \right)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}}} \\ {= {\sum\limits_{k}{p_{k}{\int{{h\left( {t - t_{k}} \right)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\;{t(22)}}}}}} \\ {{= {\sum\limits_{k}{p_{k}{{\overset{\sim}{y}}_{k}\left( {\tau - \omega} \right)}}}},(23)} \end{matrix}{{where}\text{:}}} & (21) \\ {{{\overset{\sim}{y}}_{k}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{\int{{h\left( {t - t_{k}} \right)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}} & (24) \end{matrix}$ is the elementary contribution to {tilde over (y)}(τ,ω) by event t_(k). Causal Exponential Encoding Pulse and STFT Window

In some aspects, the encoding pulse of APM and the window function of STFT may both comprise a causal exponential function. For example, h(t;ω _(h))=Θ(t)e ^(−ω) ^(h) ^(t)), and  (25) w(−t;ω _(w))=Θ(t)e ^(−ω) ^(w) ^(t),  (26) where ω_(h)≥0 and ω_(w)≥0 are inverse widths of the exponential functions.

Inserting these functions into Eq. 24 produces:

$\begin{matrix} \begin{matrix} {{{\overset{\sim}{??}}_{k}\left( {\tau,\omega} \right)} = {\int{{h\left( {t - t_{k}} \right)}{w\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t\mspace{365mu}(27)}}} \\ {= {\int{{\Theta\left( {t - t_{k}} \right)}e^{- {\omega_{h}{({t - t_{k}})}}}{\Theta\left( {\tau - t} \right)}e^{- {\omega_{w}{({\tau - t})}}}e^{{- i}\;\omega\; t}d\;{t(28)}}}} \\ {= {\int{{\Theta\left( {t - t_{k}} \right)}{\Theta\left( {\tau - t} \right)}e^{{{- \omega_{w}}\tau} + {\omega_{h}t_{k}} - {{({\omega_{h} - \omega_{w} + {i\;\omega}})}t}}d\;{t(29)}}}} \\ {= {{\Theta\left( {\tau - t_{k}} \right)}e^{{{- \omega_{w}}\tau} + {\omega_{h}t_{k}}}{\int_{t_{k}}^{\tau}{e^{{- {({\omega_{h} - \omega_{v} + {i\;\omega}})}}t}\ d\;{t(30)}}}}} \\ {= {{\Theta\left( {\tau - t_{k}} \right)}e^{{{- \omega_{w}}\tau} + {\omega_{h}t_{k}}}\frac{{- e^{{- {({\omega_{h} - \omega_{w} + {i\;\omega}})}}\tau}} + e^{{- {({\omega_{h} - \omega_{w} + {i\;\omega}})}}t_{k}}}{\omega_{h} - \omega_{w} + {i\;\omega}}(31)}} \\ {= {{\Theta\left( {\tau - t_{k}} \right)}\frac{e^{{{- \omega_{w}}\tau} + {\omega_{h}t_{k}}}}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}}\left\{ {e^{{{- {({\omega_{h} - \omega_{w}})}}\tau} - {i{\lbrack{{\omega\tau} + {\gamma{(\omega)}}}\rbrack}}} + \mspace{79mu}(32)} \right.}} \\ \left. e^{{{- {({\omega_{h} - \omega_{w}})}}t_{k}} - {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}} \right\} \\ {= {\frac{\Theta\left( {\tau - t_{k}} \right)}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}}\left\{ {e^{{- {\omega_{h}{({\tau - t_{k}})}}} - {i{\lbrack{{\omega\tau} + {\gamma{(\omega)}} - \pi}\rbrack}}} + \mspace{40mu}(33)} \right.}} \\ {\left. e^{{- {\omega_{w}{({\tau - t_{k}})}}} - {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}} \right\}.} \end{matrix} & \; \end{matrix}$ Here:

$\begin{matrix} {{\gamma\left( {{\omega;\omega_{h}},\omega_{w}} \right)}\overset{\Delta}{=}{\tan^{- 1}{\frac{\omega}{\omega_{h} - \omega_{w}}.}}} & (34) \end{matrix}$

Decomposing {tilde over (y)}_(k)(τ,ω) into two components, {tilde over (y)} _(k)(τ,ω)=ũ _(k)(τ,ω)+{tilde over (υ)}_(k)(τ,ω),  (35) where:

$\begin{matrix} {{{{\overset{\sim}{u}}_{k}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{{\Theta\left( {\tau - t_{k}} \right)}\frac{e^{{- {\omega_{h}{({\tau - t_{k}})}}} - {i{\lbrack{{\omega\;\tau} + {\gamma{(\omega)}} - \pi}\rbrack}}}}{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}}}}},{and}} & (36) \\ {{{\overset{\sim}{\upsilon}}_{k}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{{\Theta\left( {\tau - t_{k}} \right)}{\frac{e^{{- {\omega_{??}{({\tau - t_{k}})}}} - {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}}}{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}}}.}}} & (37) \end{matrix}$

The amplitude of both ũ_(k)(τ,ω) and {tilde over (υ)}_(k)(τ,ω) decay exponentially from τ=t_(k),

$\begin{matrix} {{{{{\overset{\sim}{u}}_{k}\left( {\tau - \omega} \right)}} = {{\Theta\left( {\tau - t_{k}} \right)}\frac{e^{- {\omega_{h}{({\tau - t_{k}})}}}}{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}}}}},} & (38) \\ {{{{\overset{\sim}{\upsilon}}_{k}\left( {\tau,\omega} \right)}} = {{\Theta\left( {\tau - t_{k}} \right)}{\frac{e^{- {\omega_{??}{({\tau - t_{k}})}}}}{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}}}.}}} & (39) \end{matrix}$

The instantaneous phase of ũ_(k)(τ,ω) only depends on absolute time T, and is independent of event time t_(k), arg{tilde over (υ)} _(k)(τ,ω)=ωτ−γ(ω)+π,  (40) whereas the phase of {tilde over (υ)}_(k)(τ,ω) is a constant determined by event time t_(k), independent of absolute time τ, arg {tilde over (υ)} _(k)(τ,ω)=−ωt _(k)−γ(ω).  (41)

Accordingly, defining state variable ũ(τ,ω) and {tilde over (υ)}(τ,ω) as follows:

$\begin{matrix} {{{\overset{\sim}{u}\left( {\tau,\omega} \right)} = {\sum\limits_{k}{p_{k}{{\overset{\sim}{u}}_{k}\left( {\tau,\omega} \right)}}}},{and}} & (42) \\ {{{\overset{\sim}{\upsilon}\left( {\tau,\omega} \right)} = {\sum\limits_{k}{p_{k}k^{\overset{\sim}{\upsilon}}{k\left( {\tau,\omega} \right)}}}},} & (43) \end{matrix}$ {tilde over (y)}(τ,ω) may be expressed as: {tilde over (y)}(τ,ω)={tilde over (u)}(τ,ω)+{tilde over (υ)}(τ,ω).  (44) An Event-Driven STFT Technique with Exponential Window Function and Exponential APM Encoding Pulse

An event-based technique that computes {tilde over (y)}(τ,ω) may be derived by analytically tracking the evolution of {tilde over (y)}(τ,ω) during τϵ(t_(k-1),t_(k)], for example, since event τ=t_(k-1) until the arrival of event τ=t_(k).

At τ=t_(k-1)+0 (here 0 may represent an infinitesimal positive value),

$\begin{matrix} \begin{matrix} {{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)} = {\sum\limits_{l = 1}^{k - 1}{p_{l}{{\overset{\sim}{u}}_{l}\left( {t_{k - 1},\omega} \right)}}}} \\ {= {\sum\limits_{l - 1}^{k - 1}{p_{l}{\Theta\left( {t_{k - 1} - t_{l}} \right)}\frac{e^{{- {\omega_{h}{({t_{k - 1} - t_{l}})}}} - {i{\lbrack{{\omega\; t_{k}} - 1 + {\gamma{(\omega)}} - \pi}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}}(46)}}} \\ {= {{\left\lbrack {\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}} \right\rbrack^{- \frac{1}{2}}\left\lbrack {\sum\limits_{l = 1}^{k - 1}{p_{l}e^{- {\omega_{h}{({t_{k - 1} - t_{l}})}}}}} \right\rbrack}(47)}} \\ {e^{- {i{\lbrack{{w\; t_{k - 1}} + {\gamma{(\omega)}} - \pi}\rbrack}}};} \end{matrix} & (45) \\ \begin{matrix} {{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)} = {\sum\limits_{l = 1}^{k - 1}{{??}_{l}{{\overset{\sim}{u}}_{l}\left( {t_{k - 1},\omega} \right)}}}} \\ {= {\sum\limits_{l - 1}^{k - 1}{p_{l}{\Theta\left( {t_{k - 1} - t_{l}} \right)}\frac{e^{{- {\omega_{??}{({t_{k - 1} - t_{l}})}}} - {i{\lbrack{{\omega\; t_{l}} + {\gamma{(\omega)}}}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}}(49)}}} \\ {= {\left\lbrack {\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}} \right\rbrack^{- \frac{1}{2}}{\sum\limits_{l = 1}^{k - 1}{p_{l}{e^{{- {\omega_{??}{({t_{k - 1} - t_{l}})}}} - {{i{\lbrack{{\omega\; t_{l}} + {\gamma{(\omega)}}}\rbrack}}.}}(50)}}}}} \end{matrix} & (48) \end{matrix}$

At τ=t_(k)·−0, the amplitudes of both ũ and {tilde over (υ)} may exponentially decay by the constant e^(−ω) ^(h) ^((t) ^(k) ^(-t) ^(k-1) ⁾ and

(t_(k)-t_(k-1)), whereas the phase of the former may continue its course of change and that of the latter remains a constant. This may be observed from

$\begin{matrix} \begin{matrix} {{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} = {\sum\limits_{l = 1}^{k - 1}{p_{l}{{\overset{\sim}{u}}_{l}\left( {t_{k},\omega} \right)}}}} \\ {= {\sum\limits_{l - 1}^{k - 1}{p_{l}{\Theta\left( {t_{k} - t_{l}} \right)}\frac{e^{{- {\omega_{h}{({t_{k} - t_{l}})}}} - {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}} - \pi}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}}(52)}}} \\ {= {{\left\lbrack {\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}} \right\rbrack^{- \frac{1}{2}}\left\lbrack {\sum\limits_{l = 1}^{k - 1}{p_{l}e^{- {\omega_{h}{({t_{k} - t_{l}})}}}}} \right\rbrack}{e^{i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}} - \pi}\rbrack}}(53)}}} \\ {= {{e^{{- {i{({\omega_{h} + {i\;\omega}})}}}{({t_{k} - t_{k - 1}})}}\left\lbrack {\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}} \right\rbrack}^{- \frac{1}{2}}(54)}} \\ {\left\lbrack {\sum\limits_{l = 1}^{k - 1}{p_{l}e^{- {\omega_{h}{({t_{k} - t_{l}})}}}}} \right\rbrack e^{- {i{\lbrack{{\omega\; t_{k}} - 1 + {\gamma{(\omega)}} - \pi}\rbrack}}}} \\ {{= {e^{{- {({\omega_{h} + {i\;\omega}})}}{({t_{k} - t_{k - 1}})}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}}},(55)} \end{matrix} & (51) \\ {\mspace{79mu}{{and}\mspace{14mu}{from}}} & \; \\ \begin{matrix} {{\overset{\sim}{\upsilon}\left( {{t_{k} - 0},\omega} \right)} = {\sum\limits_{l = 1}^{k - 1}{p_{l}{{\overset{\sim}{\upsilon}}_{l}\left( {t_{k},\omega} \right)}}}} \\ {= {\sum\limits_{l - 1}^{k - 1}{p_{l}{\Theta\left( {t_{k} - t_{l}} \right)}\frac{e^{{- {\omega_{??}{({t_{k} - t_{l}})}}} - {i{\lbrack{{\omega\; t_{i}} + {\gamma{(\omega)}}}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}}(57)}}} \\ {= {\left\lbrack {\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}} \right\rbrack^{- \frac{1}{2}}{\sum\limits_{l = 1}^{k - 1}{p_{l}{{e^{- {\omega_{??}{({t_{k} - t_{l}})}}}}^{- {i{\lbrack{{\omega\; t_{l}} + {\gamma{(\omega)}}}\rbrack}}}(58)}}}}} \\ {= {{e^{- {\omega_{??}{({t_{k} - t_{k - 1}})}}}\left\lbrack {\left( {\omega_{h} - \omega_{??}} \right)^{2} + \omega^{2}} \right\rbrack}^{- \frac{1}{2}}(59)}} \\ {\sum\limits_{l = 1}^{k - 1}{p_{l}{e^{- {\omega_{??}{({t_{k - 1} - t_{l}})}}}}^{- {i{\lbrack{{\omega\; t_{l}} + {\gamma{(\omega)}}}\rbrack}}}}} \\ {= {e^{- {\omega_{??}{({t_{k} - t_{k - 1}})}}}{{\overset{\sim}{\upsilon}\left( {{t_{k - 1} + 0},\omega} \right)}.(60)}}} \end{matrix} & (56) \end{matrix}$

Finally, at τ=t_(k)+0, both ũ and {tilde over (υ)} experience an increment triggered by event t_(k):

$\begin{matrix} \begin{matrix} {{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} = {\sum\limits_{l = 1}^{k}{p_{l}{{\overset{\sim}{u}}_{l}\left( {t_{k},\omega} \right)}}}} \\ {= {{p_{k}{{\overset{\sim}{u}}_{k}\left( {t_{k},\omega} \right)}} + {\sum\limits_{l = 1}^{k - 1}{p_{l}{{\overset{\sim}{u}}_{l}\left( {t_{k},\omega} \right)}(62)}}}} \\ {{= {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}} - \pi}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}} + {\overset{\sim}{u}\left( {{t_{k} - 0},\omega} \right)}}};(63)} \end{matrix} & (61) \\ \begin{matrix} {{\overset{\sim}{\upsilon}\left( {{t_{k} + 0},\omega} \right)} = {\sum\limits_{l = 1}^{k}{p_{l}{{\overset{\sim}{\upsilon}}_{l}\left( {t_{k},\omega} \right)}}}} \\ {= {{p_{k}{{\overset{\sim}{\upsilon}}_{k}\left( {t_{k},\omega} \right)}} + {\sum\limits_{l = 1}^{k - 1}{p_{l}{{\overset{\sim}{\upsilon}}_{l}\left( {t_{k},\omega} \right)}(65)}}}} \\ {= {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}} + {{\overset{\sim}{\upsilon}\left( {{t_{k} - 0},\omega} \right)}.(66)}}} \end{matrix} & (64) \end{matrix}$

Thus, the change in ũ and {tilde over (υ)} from event t_(k-1) to event t_(k) are

$\begin{matrix} {{{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}} - \pi}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}} + {e^{{- {({\omega_{h} - {i\;\omega}})}}{({t_{k} - t_{k - 1}})}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}}}};} & (67) \\ {{\overset{\sim}{\upsilon}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}} + {e^{- {\omega_{??}{({t_{k} - t_{k - 1}})}}}{{\overset{\sim}{\upsilon}\left( {{t_{k - 1} + 0},\omega} \right)}.}}}} & (68) \end{matrix}$

In summary, given a polarized event train ξ(t;{(t_(k),p_(k))})=

_(h)[x](t) as an APM sampled signal x(t) under exponential kernel function h(t)=Θ(t)e^(−w) ^(h) ^(t), the pseudo code in Table 1 describes a purely event-driven technique that computes the STFT of the direct reconstruction y(t)=ρ_(h)[ξ](t) under an exponential window function w(−t)=Θ(t)

.

Adding Eqns. 67 and 68, produces

$\begin{matrix} \begin{matrix} {{\overset{\sim}{y}\left( {{t_{k} + 0},\omega} \right)} = {{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} + {\overset{\sim}{\upsilon}\left( {{t_{k} + 0},\omega} \right)}}} \\ {= {{e^{{- {({\omega_{h} + {i\;\omega}})}}{({t_{k} - t_{k - 1}})}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}} +}} \\ {e^{- {\omega_{??}{({t_{k} - t_{k - 1}})}}}{{\overset{\sim}{\upsilon}\left( {{t_{k - 1} + 0},\omega} \right)}.}} \end{matrix} & \begin{matrix} (69) \\ (70) \end{matrix} \end{matrix}$

Combined with Eqns. 67 and 68, this particular technique results in a generalized LTI IIR filter (Eqn. 18), e.g.,

$\begin{matrix} {{\begin{bmatrix} {\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} \\ {\overset{\sim}{\upsilon}\left( {{t_{k} + 0},\omega} \right)} \\ {\overset{\sim}{y}\left( {{t_{k} + 0},\omega} \right)} \end{bmatrix} = {{{A(\omega)}{\varphi\left( {t_{k},p_{k},\omega} \right)}} + {{B\left( {{t_{k} - t_{k - 1}},\omega} \right)}\begin{bmatrix} {\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)} \\ {\overset{\sim}{\upsilon}\left( {{t_{k - 1} + 0},\omega} \right)} \\ {\overset{\sim}{y}\left( {{t_{k - 1} + 0},\omega} \right)} \end{bmatrix}}}},\mspace{20mu}{where}} & (71) \\ {\mspace{79mu}{{{\varphi\left( {t,p,\omega} \right)} = {p\; e^{{- i}\;\omega\; t}}},}} & (72) \\ {\mspace{79mu}{{{A(\omega)} = \begin{bmatrix} {- \frac{e^{{- i}\;{\gamma{(\omega)}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}}} \\ \frac{e^{{- i}\;{\gamma{(\omega)}}}}{{\sqrt{\left( {\omega_{h} - \omega_{??}} \right)}}^{2} + \omega^{2}} \\ 0 \end{bmatrix}},{and}}} & (73) \\ {\mspace{79mu}{{B\left( {{\Delta\; t},\omega} \right)} = \begin{bmatrix} e^{{- {({\omega_{h} + {i\;\omega}})}}\Delta\; t} & 0 & 0 \\ 0 & e^{{- \omega_{w}}\Delta\; t} & 0 \\ e^{{- {({\omega_{h} + {i\;\omega}})}}\Delta\; t} & e^{{- \omega_{??}}\Delta\; t} & 0 \end{bmatrix}}} & (74) \end{matrix}$

Tables 1-6 include exemplary pseudo code for processing asynchronous event driven samples of a continuous time signal in accordance with aspects of the present disclosure. As indicated in Table 1, for each event of the event time vector t and each frequency in a time window, state variable vectors u and v may be updated and used to calculate the output signal y.

TABLE 1 Pseudocode Comment 1 (t, t_(last)) ← (t₀, t₀) Current and last event times 2 t ← [t₁, . . . , t_(K)] Event time vector 3 p ← [p₁, . . . , p_(K)] Event polarity vector 4 ω ← [ω₁, . . . , ω_(L)] Frequency vector 5 $\left. \gamma\leftarrow{\tan^{- 1}\frac{\omega}{\omega_{h} - \omega_{w}}} \right.$ Phase offset vector 6 (u,v,y)←(0,0,0) State variable vectors of length L 7 for k from 1 to K do For each event 8  t ← t_(k) Current event time 9  for l from 1 to L do For each frequency 10   (ω, γ) ← (ω_(l), γ_(l)) Current frequency 11    $\left. u_{l}\leftarrow{\frac{p_{K}e^{- {i{\lbrack{{\omega\; t} + {\gamma{(\omega)}} - \pi}\rbrack}}}}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}} + {e^{{- {({\omega_{h} + {i\;\omega}})}}{({t - t_{last}})}}u_{l}}} \right.$ Update u 12    $\left. v_{l}\leftarrow{\frac{p_{K}e^{- {i{({{\omega\; t} + \gamma})}}}}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}} + {e^{- {\omega_{\omega}{({t - t_{last}})}}}v_{l}}} \right.$ Update v 13  end for 14  y←u+v Update y 15  t_(last) ← t Current event becomes last event 16 end for Matching Exponential Encoding Pulse and STFT Window

In some aspects, the exponential encoding pulse of APM and the exponential window function of STFT may be symmetric (e.g., ω_(h)=ω_(ω)=ω₀). As such the APM kernel function h(t) may be expressed as: h(t)=w(−t)=Θ(t)e ^(−ω) ⁰ ^(t).  (75)

In this case,

$\begin{matrix} {{\gamma\left( {\left. \omega \middle| \omega_{h} \right. = \omega_{w}} \right)} = {\frac{\pi}{2}.}} & (76) \end{matrix}$

The event-driven updates of the state variables ũ(τ,ω) (Eqn. 67) and of {tilde over (υ)}(τ,ω) (Eqn. 68) may thus be expressed as:

$\begin{matrix} {{{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}}{\omega} + e^{- {i{({{\omega\; t_{k}} - \frac{\pi}{2}})}}} + {e^{{- {({\omega_{0} + {i\;\omega}})}}{({t_{k} - t_{k - 1}})}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}}}};} & (77) \\ {\mspace{79mu}{{\overset{\sim}{\upsilon}\left( {{t_{k} + 0},\omega} \right)} = {{\frac{p_{k}}{\omega}e^{- {i{({{\omega\; t_{k}} + \frac{\pi}{2}})}}}} + {e^{- {\omega_{0}{({t_{k} - t_{k - 1}})}}}{{\overset{\sim}{\upsilon}\left( {{t_{k - 1} + 0},\omega} \right)}.}}}}} & (78) \end{matrix}$

Table 2 includes exemplary pseudo code for processing asynchronous event driven samples of a continuous time signal in accordance with aspects of the present disclosure. The exemplary pseudo code of Table 2 provides a simplification over the pseudo code of Table 1, as the exponential encoding pulse of APM and the exponential window function of STFT are symmetric (e.g., ω_(h)=ω_(ω)=ω₀). This may be useful for example in processing inputs having non-causal pulse shapes, which may otherwise be truncated or windowed due to potentially large delays.

TABLE 2 Pseudocode Comment 1 (t, t_(last)) ← (t₀, t₀) Current and last event times 2 t ← [t₁, . . . , t_(K)] Event time vector 3 p ← [p₁, . . . , p_(K)] Event polarity vector 4 ω ← [ω₁, . . . , ω_(L)] Frequency vector 5 (u, v, y)←(0,0,0) State variable vectors of length L 6 for k from 1 to K do For each event 7  t ← t_(k) Current event time 8  for l from 1 to L do For each frequency 9   ω ← ω_(l) Current frequency 10    $\left. u_{l}\leftarrow{{\frac{p_{k}}{\omega}e^{- {i{({{\omega\; t} - \frac{\pi}{2}})}}}} + {e^{{- {({\omega_{0} + {i\;\omega}})}}{({t - t_{last}})}}u_{l}}} \right.$ Update u 11    $\left. v_{l}\leftarrow{{\frac{p_{k}}{\omega}e^{- {i{({{\omega\; t} + \frac{\pi}{2}})}}}} + {e^{- {\omega_{0}{({t - t_{last}})}}}v_{l}}} \right.$ Update v 12  end for 13  y←u+v Update y 14  t_(last) ← t Current event becomes last event 15 end for Level-Crossed Sampling and Exponential STFT Window

In some aspects, the input signal may be subjected to level-crossed sampling (e.g., APM sampled under the heavy-side step function as encoding pulse). In this case ω_(h) may be set as: ω_(h)=0,  (79) and ω=w=ω₀. Further, phase offset γ may be given by:

$\begin{matrix} {{\gamma\left( {{\omega;0},\omega_{0}} \right)} = {\tan^{- 1}\left( {- \frac{\omega}{\omega_{0}}} \right)}} & (80) \end{matrix}$

Accordingly, the event-driven updates of ũ(τ,ω) (Eqn. 67) and of {tilde over (υ)}(τ,ω) (Eqn. 68) may be expressed as:

$\begin{matrix} {{{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}} - \pi}\rbrack}}}}{\sqrt{\omega_{0}^{2} + \omega^{2}}} + {e^{{- i}\;{\omega{({t_{k} - t_{k - 1}})}}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}}}};} & (81) \\ {{\overset{\sim}{\upsilon}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}}}{\sqrt{\omega_{0}^{2} + \omega^{2}}} + {e^{- {\omega_{w}{({t_{k} - t_{k - 1}})}}}{{\overset{\sim}{\upsilon}\left( {{t_{k - 1} + 0},\omega} \right)}.}}}} & (82) \end{matrix}$

Table 3 includes exemplary pseudo code for processing asynchronous event driven samples of a continuous time signal in accordance with aspects of the present disclosure. In the exemplary pseudo code of Table 3, the pseudo code of Table 1 may be simplified where the input signal is sampled using a level-crossed sampling method (e.g., Lebesgue sampling). For example, in one aspect, the input may be APM sampled under the heavy-side step function as the encoding pulse. By applying the heavy-side function, the pulse becomes causal. As such, the input may be processed with much less delay (e.g., real time processing).

TABLE 3 Pseudocode Comment 1 (t, t_(last)) ← (t₀, t₀) Current and last event times 2 t ← [t₁, . . . , t_(k)] Event time vector 3 p ← [p₁, . . . , p_(K)] Event polarity vector 4 ω ← [ω₁, . . . , ω_(L)] Frequency vector 5 $\left. \gamma\leftarrow{\tan^{- 1} - \frac{\omega}{\omega_{0}}} \right.$ Phase offset vector 6 (u, v, y)←(0,0,0) State variable vectors of length L 7 for k from 1 to K do For each event 8  t ← t_(k) Current event time 9  for l from 1 to L do For each frequency 10   (ω, γ) ← ω_(l), γ_(l) Current frequency 11    $\left. u_{l}\leftarrow{\frac{p_{k}e^{- {i{({{\omega\; t} + \gamma - \pi})}}}}{\sqrt{\omega_{0}^{2} + \omega^{2}}} + {e^{{- i}\;{\omega{({t - t_{last}})}}}u_{l}}} \right.$ Update u 12    $\left. v_{l}\leftarrow{\frac{p_{k}e^{- {i{({{\omega\; t} + \gamma})}}}}{\sqrt{\omega_{0}^{2} + \omega^{2}}} + {e^{- {\omega_{0}{({t - t_{last}})}}}v_{l}}} \right.$ Update v 13  end for 14  y←u+v Update y 15  t_(last) ← t Current event becomes last event 16 end for Generalization: Complex Exponential Sum APM Encoding Kernels and STFT Window Functions

In some aspect, the process may be generalized such that the APM kernel h(t) and STFT window w(t) can be any arbitrary causal functions.

Complex ω_(h) and ω_(ω),

In one example, the event-driven technique described in Table 1 may be generalized to the case where ω_(h) and ω_(ω)are complex, e.g., ω_(h)=α_(h) +iβ _(h) and  (83) ω_(ω)=α_(ω) +iβ _(ω),  (84) but with non-negative real parts, e.g., α_(h),

≥0.

Thus, the contribution to {tilde over (y)}(τ,ω) by event t_(k) (Eqn. 24) becomes

k ⁢ ( τ , ω ) = ⁢ ∫ h ⁡ ( t - t k ) ⁢ w ⁡ ( t - τ ) ⁢ e - i ⁢ ⁢ ω ⁢ ⁢ t ⁢ d ⁢ ⁢ t ( 85 ) ⁢ = ⁢ ∫ Θ ⁡ ( t - t k ) ⁢ e - ω h ⁡ ( t - t k ) ⁢ Θ ⁡ ( τ - t ) ⁢ e - ω w ( τ - t ) ⁢ e - i ⁢ ⁢ ω ⁢ ⁢ t ⁢ d ⁢ ⁢ t ( 86 ) ⁢ = ⁢ ∫ Θ ⁡ ( t - t k ) ⁢ Θ ⁡ ( τ - t ) ⁢ e - ω w ⁢ τ + ω h ⁢ t k - ⁡ ( ω h - ⁢ ω w + i ⁢ ⁢ ω ) ⁢ t ⁢ d ⁢ ⁢ t ( 87 ) ⁢ = ⁢ Θ ⁡ ( τ - t k ) ⁢ e - ω w ⁢ τ + ω h ⁢ t k ⁢ ∫ t k τ ⁢ e - ( ω h - ω w + i ⁢ ⁢ ω ) t ⁢d ⁢ ⁢ t ( 88 ) ⁢ = ⁢ Θ ⁡ ( τ - t k ) ⁢ e - ω w ⁢ τ + ω h ⁢ t k ⁢ - e - e ⁡ ( ω h - ω w + i ⁢ ⁢ ω ) ⁢ τ + e - ( ω h - ω w + i ⁢ ⁢ ω ) ⁢ t k ω h - ω w + i ⁢ ⁢ ω ( 89 ) ⁢ = ⁢ Θ ⁡ ( τ - t k ) ⁢ e - ( α w + i ⁢ ⁢ βω ) ⁢ τ + ( α h + i ⁢ ⁢ β h ) ⁢ t k ⁢ - e - [ α h - α w + i ⁡ ( β h - β w + ω ) ] ⁢ τ + e - [ α h - α w + i ⁡ ( β h - β w + ω ) ] ⁢ t k α h - α w + i ⁡ ( β h - β w + ω ) = ⁢ Θ ⁡ ( τ - t k ) ⁢ e - ( α w + i ⁢ ⁢ β w ) ⁢ τ + ( α h + i ⁢ ⁢ β h ) ⁢ t k ( α h - α w ) 2 + ( β h - β w + ω ) 2 ( 90 ) ⁢ = ⁢ { - e - ( α h - α w ) ⁢ τ - i ⁡ [ ( β h - β w + ω ) ⁢ τ + γ ⁡ ( ω ) ] + e - ( α h - α w ) ⁢ t k - i [ β h - β w ) ⁢ t k + γ ⁡ ( ω ) ] } ( 91 ) ⁢ = ⁢ Θ ⁡ ( τ - t k ) ( α h - α w ) 2 + ( β h - β w + ω ) 2 ⁢ { e - ( α h + i ⁢ ⁢ β h ) ⁢ ( τ - t k ) - i ⁡ [ ω ⁢ ⁢ τ + γ ⁡ ( ω ) - π ] + e ( - α w + i ⁢ ⁢ β w ) ⁢ ( τ - t k ) - i ⁡ [ wt k + γ ⁡ ( ω ) ] } , ( 92 ) where γ ⁡ ( ω ; α h , β h , α w , β w ) ⁢ = Δ ⁢ tan - 1 ⁢ β h - β w + ω α h - α w . ( 93 )

The decomposition of

_(k)(τ,ω) thus produces state variables:

$\begin{matrix} {{{{\overset{\sim}{u}}_{k}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{{\Theta\left( {\tau - t_{k}} \right)}\frac{e^{{- {({\alpha_{h} + {i\;\beta_{h}}})}}{({\tau - t_{k} - {i{\lbrack{{\omega\;\tau} + {\gamma{(\omega)}} - \pi}\rbrack}}}}}}{\sqrt{\left( {\alpha_{h} - \alpha_{w}} \right)^{2} + \left( {\beta_{n} - \beta_{w} + \omega} \right)^{2}}}}},} & (94) \\ {{{\overset{\sim}{u}}_{k}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{{\Theta\left( {\tau - t_{k}} \right)}\;{\frac{e^{{- {({\alpha_{w} + {i\;\beta_{\omega}}})}}{({\tau - t_{k} - {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}}}}}{\sqrt{\left( {\alpha_{h} - \alpha_{w}} \right)^{2} + \left( {\beta_{n} - \beta_{w} + \omega} \right)^{2}}}.}}} & (95) \end{matrix}$

Likewise, a linear recursive relationship of states if, ũ, {tilde over (v)} and {tilde over (y)} from event t_(k-1) to event t_(k) as follows:

$\begin{matrix} {{{\overset{\sim}{u}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma{(\omega)}}}\rbrack}}}}{\sqrt{\left( {\alpha_{h} - \alpha_{w}} \right)^{2} + \left( {\beta_{n} - \beta_{w} + \omega} \right)^{2}}} + {e^{{- {\lbrack{\alpha_{h} + {i{({\beta_{h} + \omega})}}}\rbrack}}{({t_{k} - t_{k}})}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}}}};} & (96) \\ {{\overset{\sim}{v}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{wt}_{k} + {\gamma{(\omega)}}}\rbrack}}}}{\sqrt{\left( {\alpha_{h} - \alpha_{w}} \right)^{2} + \left( {\beta_{n} - \beta_{w} + \omega} \right)^{2}}} + {e^{{- {({\alpha_{w} + {i\;\beta_{w}}})}}{({t_{k} - t_{k - 1}})}}{\overset{\sim}{v}\left( {{t_{k - 1} + 0},\omega} \right)}}}} & (97) \\ {{\overset{\sim}{y}\left( {{t_{k} + 0},\omega} \right)} = {{e^{{- {\lbrack{\alpha_{h} + {i{({\beta_{h} + \omega})}}}\rbrack}}{({t_{k} - t_{k - 1}})}}{\overset{\sim}{u}\left( {{t_{k - 1} + 0},\omega} \right)}} + {e^{{- {({\alpha_{w} + {i\;\beta_{w}}})}}{(t_{k - 1})}}{\overset{\sim}{v}\left( {{t_{k - 1} + 0},\omega} \right)}}}} & (98) \end{matrix}$ Complex Exponential Sum

Considering an arbitrary causal function, f(t)=Θ(t)g(t), if g(t) can be expressed as an complex exponential sum, e.g.:

$\begin{matrix} {{{g(t)} = {\sum\limits_{j}{a_{j}e^{b_{j}t}}}},} & (99) \end{matrix}$ where α_(j), b_(j)ϵ

, then f(t) may be expressed as a complex weighted sum of causal complex exponentials, e.g.:

$\begin{matrix} {{f(t)} = {{\Theta(t)}{\sum\limits_{j}{a_{j}e^{b_{j}t}}}}} & (100) \\ {\mspace{40mu}{= {\sum\limits_{j}{a_{j}\left\lbrack {{\Theta(t)}e^{b_{j}t}} \right.}}}} & (101) \\ {\mspace{40mu}{\overset{\Delta}{=}{\sum\limits_{j}{a_{j}{f_{j}\left( {t;b_{j}} \right)}}}}} & (102) \end{matrix}$

This form is quite expressive in terms of approximating arbitrary functions, (e.g., using Prony's method).

In some aspects, a real function expressed as a weighted sum of causal, damped oscillations may be converted to the form of complex weighted sum of causal complex exponentials. First, the real function as weighted sum of N causal, damped oscillations may be expressed as:

$\begin{matrix} {{{f(t)} = {\sum\limits_{n = 1}^{N}{{\Theta(t)}\rho_{n}e^{{- \sigma_{n}}t}{\cos\left( {{\omega_{n}t} + \phi_{n}} \right)}}}},} & (103) \end{matrix}$ where ρ_(n)'s, σ_(n)'s, ω_(n)'s and φ_(n)'s are the amplitudes, damping constants, angular frequencies and phases of the components.

Constructing (a_(j), b_(j))(j=1, . . . , 2N) by conjugate pairs of complex exponentials

$\begin{matrix} {a_{j} = {\frac{1}{2}\rho_{\lceil\frac{j}{2}\rceil}e^{{- {i{({- 1})}}^{j}}\varnothing_{\lceil\frac{j}{2}\rceil}}\mspace{14mu}{and}}} & (104) \\ {{b_{j} = {{- \sigma_{\lceil\frac{j}{2}\rceil}} - {{i\left( {- 1} \right)}^{j}\omega_{\lceil\frac{j}{2}\rceil}}}},{produces}} & (105) \\ {{{\Theta(t)}{\sum\limits_{j}^{2N}{a_{j}e^{b_{j}t}}}} = {{\sum\limits_{n}^{N}{{\Theta(t)}\rho_{n}e^{{- \sigma_{n}}t}{\cos\left( {{\omega_{n}t} + \varnothing_{n}} \right)}}} = {{f(t)}.}}} & (106) \end{matrix}$

As such, a real causal function in the form of Eqn. 103 can be expressed as a complex weighted sum of causal complex exponentials.

APM Kernel and STFT Window as Complex Weighted Sums of Causal Complex Exponentials

In some aspects, h(t) and w(t) may be generally expressed in a form in which both are complex weighted sums of causal complex exponentials, e.g.,

$\begin{matrix} {{h\left( {{t;a_{h}^{(1)}},\ldots\mspace{14mu},a_{h}^{(N)},b_{h}^{(1)},\ldots\mspace{14mu},b_{h}^{(N)}} \right)} = {{\sum\limits_{n = 1}^{N}{a_{h}^{(n)}{h^{(n)}\left( {t;b_{h}^{(n)}} \right)}}} = {\sum\limits_{n = 1}^{N}{a_{h}^{(n)}{\Theta(t)}e^{b_{h}^{{(n)}_{t}}}}}}} & (107) \\ {w\left( {{{- t};a_{w}^{(1)}},\ldots\mspace{14mu},a_{w}^{(M)},b_{w}^{(1)},\ldots\mspace{14mu},{b_{w}^{(M)} = {{\sum\limits_{m - 1}^{M}{a_{w}^{(m)}{w^{(m)}\left( {{- t};b_{w}^{(m)}} \right)}}} = {\sum\limits_{m = 1}^{M}{a_{w}^{(m)}{\Theta(t)}{e^{b_{\omega}^{{(m)}t}}.}}}}}} \right.} & (108) \end{matrix}$

Then, the contribution to {tilde over (y)}(τ,ω) by event t_(k) (Eqn. 24) becomes

$\begin{matrix} {{{\overset{\sim}{y}}_{k}\left( {\tau,\omega} \right)} = {\int{{h\left( {t - t_{k}} \right)}{w\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}} & (109) \\ \begin{matrix} {\mspace{85mu}{= {\int\left\lbrack {{\Theta\left( {t - t_{k}} \right)}{\sum\limits_{n = 1}^{N}{a_{h}^{(n)}{h^{(n)}\left( {t - t_{k}} \right)}}}} \right\rbrack}}} \\ {\left\lbrack {{\Theta\left( {\tau - t} \right)}{\sum\limits_{m = 1}^{M}{a_{w}^{(m)}{w^{(m)}\left( {\tau - t} \right)}}}} \right\rbrack e^{{- i}\;\omega\; t}d\; t} \end{matrix} & (110) \\ \begin{matrix} {\mspace{85mu}{= {\int\left\lbrack {{\Theta\left( {t - t_{k}} \right)}{\Theta\left( {\tau - t} \right)}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{a_{h}^{(n)}a_{w}^{(m)}{h^{(n)}\left( {t - t_{k}} \right\rbrack}}}}} \right.}}} \\ {{w^{(m)}\left( {\tau - t} \right)}e^{{- i}\;\omega\; t}d\; t} \end{matrix} & (111) \\ \begin{matrix} {{= {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{a_{h}^{(n)}a_{w}^{(m)}{\int{{\Theta\left( {t - t_{k}} \right)}{\Theta\left( {\tau - t} \right)}{h^{(n)}\left( {t - t_{k}} \right)}w^{(m)}}}}}}}} \\ {\left( {\tau - t} \right)e^{{- i}\;\omega\; t}d\; t} \end{matrix} & (112) \\ {\mspace{85mu}{{= {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{a_{h}^{(n)}a_{w}^{(m)}{\overset{\sim}{y}}_{k}^{({n,m})}\left( {\tau,\omega} \right)}}}},}} & (113) \\ {where} & \; \\ {{{\overset{\sim}{y}}_{k}^{({n,m})}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{\int{{\Theta\left( {t - t_{k}} \right)}{\Theta\left( {\tau - t} \right)}{h^{(n)}\left( {t - t_{k}} \right)}{w^{(m)}\left( {\tau - t} \right)}e^{{- i}\;\omega\; t}d\; t}}} & (114) \\ \begin{matrix} {\mspace{121mu}{= {{\int{{\Theta\left( {t - t_{k}} \right)}{\Theta\left( {\tau - t} \right)}{e^{b_{h}^{(n)}}\left( {t - t_{k}} \right)}}} +}}} \\ {{b_{w}^{(m)}\left( {\tau - t} \right)}^{{- i}\;\omega\; t}d\; t} \end{matrix} & (115) \end{matrix}$ Writing b _(h) ^((n))=−(α_(n) ^((n)) +iβ _(h) ^((n)))  (116) b _(w) ^((m))=−(α_(w) ^((m)) +iβ _(w) ^((m))),  (117) {tilde over (y)}_(k) ^((n,m))(τ,ω) may be further decomposed and expressed as:  (118)

$\begin{matrix} {\mspace{20mu}{{{{\overset{\sim}{y}}_{k}^{({n,m})}\left( {\tau,\omega} \right)} = {{{\overset{\sim}{u}}_{k}^{({n,m})}\left( {\tau,\omega} \right)} + {{\overset{\sim}{v}}_{k}^{({n,m})}\left( {\tau,\omega} \right)}}},\mspace{20mu}{where}}} & (119) \\ {{{{\overset{\sim}{u}}_{k}^{({n,m})}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{{\Theta\left( {\tau - t_{k}} \right)}\frac{{{e^{-}\left( {\alpha_{h}^{(n)} + {i\;\beta_{h}^{(n)}}} \right)}\left( {\tau - t_{k}} \right)} - {i\left\lbrack {{\omega\;\tau} + {\gamma^{({n,m})}(\omega)} - \pi} \right\rbrack}}{\sqrt{\left( {\alpha_{h}^{(n)} - \alpha_{w}^{(m)}} \right)^{2}} + \left( {\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega} \right)^{2}}}},\mspace{20mu}{and}} & (120) \\ {{{{\overset{\sim}{\upsilon}}_{k\;}^{({n,m})}\left( {\tau,\omega} \right)}\overset{\Delta}{=}{{\Theta\left( {\tau - t_{k}} \right)}\frac{{{e^{-}\left( {\alpha_{h}^{(n)} + {i\;\beta_{h}^{(n)}}} \right)}\left( {\tau - t_{k}} \right)} - {i\left\lbrack {{\omega\; t_{k}} + {\gamma^{({n,m})}(\omega)}} \right\rbrack}}{\sqrt{\left( {\alpha_{h}^{(n)} - \alpha_{w}^{(m)}} \right)^{2}} + \left( {\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega} \right)^{2}}}},\mspace{20mu}{{and}\mspace{20mu}{with}}} & (121) \\ {\mspace{20mu}{{\gamma^{({n,m})}(\omega)}\overset{\Delta}{=}{\tan^{- 1}{\frac{\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega}{\alpha_{h}^{(n)} - \alpha_{w}^{(m)}}.}}}} & (122) \end{matrix}$

Defining states

$\begin{matrix} {{{\overset{\sim}{u}}^{({n,m})}\left( {\tau,\omega} \right)} = {\sum\limits_{k}{p_{k}{{\overset{\sim}{u}}_{k}^{({n,m})}\left( {\tau,\omega} \right)}\mspace{14mu}{and}}}} & (123) \\ {{{{\overset{\sim}{\upsilon}}^{({n,m})}\left( {\tau,\omega} \right)} = {\sum\limits_{k}{p_{k}{{\overset{\sim}{v}}_{k}^{({n,m})}\left( {\tau,\omega} \right)}}}},{produces}} & (124) \\ {{\overset{\sim}{y}\left( {\tau,\omega} \right)} = {\sum\limits_{n}{\sum\limits_{m}{a_{h}^{(n)}{{a_{w}^{(m)}\left\lbrack {{{\overset{\sim}{u}}^{({n,m})}\left( {\tau,\omega} \right)} + {{\overset{\sim}{\upsilon}}^{({n,m})}\left( {\tau,\omega} \right)}} \right\rbrack}.}}}}} & (125) \end{matrix}$

Because the form of ũ^((n,m)) (τ,ω) and {tilde over (υ)}(n,m) (τ,ω) is essentially the same as ũ(τ,ω) and {tilde over (υ)}(τ,ω) in Eqns. 42 and 43, the linear recursive relationship of states ũ^((n,m)) {tilde over (υ)}^((n,m)) and {tilde over (y)} from event t_(k-1) to event t_(k) as follows:

$\begin{matrix} {{{{\overset{\sim}{u}}^{({n,m})}\left( {{t_{k} + 0},\omega} \right)} = {{- \frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma^{({n,m})}{(\omega)}}}\rbrack}}}}{\sqrt{\left( {\alpha_{h}^{(n)} - \alpha_{w}^{(m)}} \right)^{2} + \left( {\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega} \right)^{2}}}} + {{e^{-}\left\lbrack {\alpha_{h}^{(n)} + {i\left( {\beta_{h}^{(n)} + \omega} \right)}} \right\rbrack}\left( {t_{k} - t_{k - 1}} \right)_{{\overset{\sim}{u}}^{({n,m})}{({{t_{k - 1} + 0},\omega})}}}}};} & (126) \\ {{{{\overset{\sim}{\upsilon}}^{({n,m})}\left( {{t_{k} + 0},\omega} \right)} = {\frac{p_{k}e^{- {i{\lbrack{{\omega\; t_{k}} + {\gamma^{({n,m})}{(\omega)}}}\rbrack}}}}{\sqrt{\left( {\alpha_{h}^{(n)} - \alpha_{w}^{(m)}} \right)^{2} + \left( {\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega} \right)^{2}}} + {{e^{-}\left\lbrack {\alpha_{w}^{(m)} + {i\;\beta_{w}^{(m)}} + \omega} \right\rbrack}\left( {t_{k} - t_{k - 1}} \right)_{{\overset{\sim}{v}}^{({n,m})}{({{t_{k - 1} + 0},\omega})}}}}};} & (127) \\ {{\overset{\sim}{y}\left( {{t_{k} + 0},\omega} \right)} = {{\sum\limits_{n,m}{\alpha_{h}^{(n)}\alpha_{w}^{(m)}{e^{-}\left\lbrack {\alpha_{n}^{n} + {i\;\left( {\beta_{h}^{(n)} + \omega} \right)}} \right\rbrack}\left( {t_{k} - t_{k - 1}} \right)_{{\overset{\sim}{u}}^{({n,m})}{({{t_{k - 1} + 0},\omega})}}}} + \alpha_{h}^{(n)} - {\alpha_{w}^{(m)}{e^{-}\left( {\alpha_{w}^{(m)} + {i\;\beta_{w}^{(m)}}} \right)}\left( {t_{k} - t_{k - 1}} \right)_{{\overset{\sim}{v}}^{({n,m})}{({{t_{k - 1} + 0},\omega})}}}}} & (128) \end{matrix}$

Therefore, the event-driven STFT technique described in Table 1 can be generalized to what is summarized in Table 4.

Table 4 includes exemplary pseudo code similar to Table 1, but further generalized to allow for the APM kernel function h(t) and STFT window function w(t) as arbitrary causal functions. That is, Table 4 summarizes a generalized design to accommodate general pulse shapes (e.g., other than exponentials).

TABLE 4 Pseudocode Comment 1 (t, t_(last)) ← (t₀, t₀) Current and last event times 2 t ← [t₁, . . . , t_(k)] Event time vector 3 p ← [p₁, . . . , p_(K)] Event polarity vector 4 ω ← [ω₁, . . . , ω_(L)] Frequency vector 5 a_(h) ← [α_(h) ⁽¹⁾, . . . , α_(h) ^((N))] Complex weights 6 α_(h) ← [α_(h) ⁽¹⁾, . . . , α_(h) ^((N))] Real part of complex exponent 7 β_(h) ← [β_(h) ⁽¹⁾, . . . , β_(h) ^((N))] Imaginary part of complex exponent 8 a_(w) ← [α_(w) ⁽¹⁾, . . . , α_(h) ^((M))] Complex weights 9 α_(w) ← [α_(w) ⁽¹⁾, . . . , α_(w) ^((M))] Real part of complex exponent 10 β_(w) ← [β_(w) ⁽¹⁾, . . . , β_(h) ^((M))] Imaginary part of complex exponent 11 for n from 1 to N do 12  for mfrom 1 to M do 13    $\left. \gamma^{({n,m})}\leftarrow{\tan^{- 1}\frac{\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega}{\alpha_{h}^{(n)} - \alpha_{w}^{(m)}}} \right.$ Phase offset vector 14  (u^((n, m)), v^((n, m))) ← (0,0) State variable vectors 15  end for 16 end for 17 y ← 0 State variable vector 18 for k from 1 to K do For each event 19  t ← t_(k) Current event time 20  for n from 1 to N do 21   for m from 1 to M do 22     $\left. u^{({n,m})}\leftarrow{{- \frac{p_{k}e^{- {i{({{\omega\; t} + \gamma^{({n,m})}})}}}}{\sqrt{\left( {\alpha_{h}^{(n)} - \alpha_{w}^{(m)}} \right)^{2} + \left( {\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega} \right)^{2}}}} + {e^{{- {\lbrack{\alpha_{h}^{(n)} + {i{({\beta_{h}^{(n)} + \omega})}}}\rbrack}}{({t - t_{last}})}}u^{({n,m})}}} \right.$ Update u^((n, m)) 23     $\left. v^{({n,m})}\leftarrow{{- \frac{\left( {p_{k}e^{i{({{\omega\; t} + \gamma^{({n,m})}})}}} \right)}{\sqrt{\left( {\alpha_{h}^{(n)} - \alpha_{w}^{(m)}} \right)^{2} + \left( {\beta_{h}^{(n)} - \beta_{w}^{(m)} + \omega} \right)^{2}}}} + {e^{{- {({\alpha_{w}^{(m)} + {i\;\beta_{w}^{(m)}}})}}{({t - t_{last}})}}\; v^{({n,m})}}} \right.$ Update v^((n, m)) 24   end for 25  end for 26 y ← Σ_(n, m)a_(h) ^((n))a_(w) ^((m))[u^((n, m)) + v^((n, m))] Update y 27 t_(last) ← t Current event becomes last event 28 end for Discrete Spatial Fourier Transform

Considering a continuous spatio-temporal signal: X(t)=[x _(r)](t)=[x _(μυ)](t)  (129)

sampled by a rectangular sensor array of dimension M×N, each element (e.g., pixel) r=[μ,υ]^(T)(μ=1, . . . , M; υ=1, . . . , N) of this sensor array may be an event-based sampler of either type defined above. We denote a single-pixel signal by x_(μ,υ)(t).

Instantaneous Spatial DFT

Discrete Fourier transform (DFT) matrix of the L-th order may be the following complex matrix:

$\begin{matrix} {S_{L} = {\left\lbrack \frac{\omega_{L}^{j\; k}}{\sqrt{L}} \right\rbrack_{j,{k = 0},\ldots\mspace{14mu},{L - 1}} = \left\lbrack {s_{L}^{0},\ldots\mspace{14mu},s_{L}^{k},\ldots\mspace{14mu},s_{L}^{L - 1}} \right\rbrack}} & (130) \\ {\mspace{25mu}{{= {\frac{1}{\sqrt{L}}\begin{bmatrix} 1 & 1 & 1 & \ldots & 1 \\ 1 & \omega_{L} & \omega_{L}^{2} & \ldots & \omega_{L}^{L - 1} \\ 1 & \omega_{L}^{2} & \omega_{L}^{4} & \ldots & \omega_{L}^{2{({L - 1})}} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \omega_{L}^{L - 1} & \omega_{L}^{2{({L - 1})}} & \ldots & \omega_{L}^{L - {1{({L - 1})}}} \end{bmatrix}}},{where}}} & (131) \\ {\omega_{L}\overset{\Delta}{=}e^{- \frac{2\pi\; i}{L}}} & (132) \end{matrix}$ is the L-th root of unity and

$\begin{matrix} {s_{L}^{k}\overset{\Delta}{=}{\frac{1}{\sqrt{L}}\begin{bmatrix} 1 \\ \omega_{L}^{k} \\ \vdots \\ \omega_{L}^{k{({L - 1})}} \end{bmatrix}}} & (133) \end{matrix}$ is the (k+1)-th column of S_(L).

At time t, the instantaneous 2-dimensional spatial DFT of x(r,t) at spatial frequency vector or ω_(r)=[ω_(μ),ω_(υ)]^(T)(ω_(μ)=0, . . . , M−1; ω_(v)=0, . . . , N−1) may be defined as:

$\begin{matrix} {{\overset{\sim}{X}(t)}\overset{\Delta}{=}{\left\lbrack {\overset{\sim}{X}}_{{\omega\;\mu},\omega_{v}} \right\rbrack(t)}} & (134) \\ {\mspace{45mu}{= {\left\lbrack {S_{M}X\; S_{N}^{T}} \right\rbrack(t)}}} & (135) \\ {\mspace{45mu}{= {\sum\limits_{\mu,v}{S_{M}{x_{\mu,v}(t)}{S_{N}^{T}.}}}}} & (136) \end{matrix}$ Short-Time Spatial DFT

Short-time spatial DFT of spatio-temporal signal X(t)=[x_(μ,v)](t), denoted by {tilde over (X)}(τ)=[{tilde over (X)}_(ωμ,ω) _(v) ](τ) may be defined as: {tilde over (X)}(τ)=S _(M) X(t)S _(N) ^(T) w(t−τ)dt,  (137) where w(t) is a window function.

Accordingly, the instantaneous spatial DFT is simply a short-time spatial DFT in which the window function is

(−t)=δ(t).

Short-Time Spatial DFT of Polarized Event Trains

Let signals X(t)=[x_(μ,υ)](t) be polarized event trains {(t_(μ,υ,k′),p_(μ,υ,k′))}:

$\begin{matrix} {{x_{\mu,v}(t)} = {\sum\limits_{k^{\prime}}{p_{\mu,v,k^{\prime}}{{\delta\left( {t - t_{\mu,v,k^{\prime}}} \right)}.}}}} & (138) \end{matrix}$

We can rewrite and combine {(t_(μ,υ,k′), p_(μ,υ,k′))}∀μ, υ into {(t_(k),p_(k),μ_(k),υ_(k))} by using in essence an event-address representation (AER) scheme, where all events in the M×N signals are indexed by a single subscript k, while the identity of each event (μ_(k),v_(k)) is attached to its time stamp and polarity. Thus, Eqn. 138 can be rewritten as

$\begin{matrix} {{{x_{\mu,v}(t)} = {\sum\limits_{k}{p_{k}\delta_{\mu,\mu_{k}}\delta_{v,v_{k}}{\delta\left( {t - t_{k}} \right)}}}};} & (139) \end{matrix}$

Thus,

$\begin{matrix} {{{X(t)} = {\sum\limits_{k}{p_{k}\Delta_{\mu_{k},v_{k}}{\delta\left( {t - t_{k}} \right)}}}},} & (140) \end{matrix}$ where Δ_(μ) _(k,) _(ν) _(k) =[δ_(μ,μ) _(k) δ_(ν,ν) _(k) ]  (141) is a discrete M×N matrix with 1 at the μ_(k)-th row and ν_(k)-th column and 0 elsewhere.

Inserting into Eqn. 137, produces

$\begin{matrix} {{\overset{\sim}{X}(\tau)} = {\int{S_{M}{\sum\limits_{k}{p_{k}\Delta_{u_{k},v_{k}}{\delta\left( {t - t_{k}} \right)}S_{N}^{T}{\omega\left( {t - \tau} \right)}d\; t}}}}} & (142) \\ {\mspace{50mu}{= {\sum\limits_{k}{\int{{p_{k}\left\lbrack {S_{M}\Delta_{\mu_{k},v_{k}}S_{N}} \right\rbrack}{\delta\left( {t - {t_{k)}{\omega\left( {t - \tau} \right)}d\; t}} \right.}}}}}} & (143) \\ {\mspace{50mu}{= {\sum\limits_{k}{\int{{p_{k}\left\lbrack {S_{M}^{\mu_{k} - 1}S_{N}^{v_{k} - 1^{T}}} \right\rbrack}{\delta\left( {t - {t_{k)}{\omega\left( {t - \tau} \right)}d\; t}} \right.}}}}}} & (144) \\ {\mspace{50mu}{= {\sum\limits_{k}{{p_{k}\left\lbrack {S_{M}^{\mu_{k} - 1}S_{N}^{v_{k} - 1^{T}}} \right\rbrack}{\int{\delta\left( {t - {t_{k)}{\omega\left( {t - \tau} \right)}d\; t}} \right.}}}}}} & (145) \\ {\mspace{50mu}{{= {\sum\limits_{k}{{p_{k}\left\lbrack {S_{M}^{\mu_{k} - 1}S_{N}^{v_{k} - 1^{T}}} \right\rbrack}{\omega\left( {t_{k} - \tau} \right)}}}},}} & (146) \end{matrix}$ where matrix [S_(M) ^(μ) ^(k) S_(N) ^(ν) ^(k) ⁻¹ ^(T) ] is the outer product between the μ_(k)-th column of S_(M) and the column of S_(N).

Defining {tilde over (X)} _(k)(τ)=[S _(M) ^(μ) ^(k) S _(N) ^(ν) ^(k) ⁻¹ ^(T) ]w(t _(k)−τ),  (147) {tilde over (X)}(τ) may be expressed as:

$\begin{matrix} {{\overset{\sim}{X}(\tau)} = {\sum\limits_{k}{p_{k}{{{\overset{\sim}{X}}_{k}(\tau)}.}}}} & (148) \end{matrix}$ Event-Driven Exponential Windowed Short-Time Spatial DFT of Polarized Event Trains

Given a causal exponential window function w(−t)=Θ(t)e^(−ω) ^(ω) ^(t), {tilde over (X)}_(k) may be expressed as: {tilde over (X)} _(k)(τ)=[s _(M) ^(μ) ^(k) ⁻¹ s _(N) ^(ν) ^(k) ⁻¹ ^(T) ]Θ(τ−t _(k))e ^(−ω) ^(ω) ^((τ-t) ^(k) ⁾,  (149) and thus,

$\begin{matrix} {{{\overset{\sim}{X}}_{k}(\tau)} = {\sum\limits_{k}{{p_{k}\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack}{\Theta\left( {\tau - t_{k}} \right)}{e^{- {\omega_{w}{({\tau - t_{k}})}}}.}}}} & (150) \end{matrix}$

An event-based technique that computes {tilde over (X)}(τ), by tracking analytically the evolution of {tilde over (X)}(τ) during τϵ(t_(k-1),t_(k)] e.g., since event τ=t_(k-1) until the arrival of event τ=t_(k) may be derived.

Immediately after event τ=t_(k-1),

$\begin{matrix} {{{\overset{\sim}{X}\left( {t_{k - 1} + 0} \right)} = {\sum\limits_{l = 1}^{k - 1}{{p_{l}\left\lbrack {s_{M}^{\mu_{l} - 1}s_{N}^{v_{l} - 1^{T}}} \right\rbrack}e^{- {\omega_{w}{({\tau - t_{l}})}}}}}};} & (151) \end{matrix}$ immediately before the arrival of event τ=t_(k),

$\begin{matrix} {{\overset{\sim}{X}\left( {t_{k} - 0} \right)} = {\sum\limits_{l = 1}^{k - 1}{{p_{l}\left\lbrack {s_{M}^{\mu_{l} - 1}s_{N}^{v_{l} - 1^{T}}} \right\rbrack}e^{- {\omega_{w}{({\tau - t_{l}})}}}}}} & (152) \\ {{= {e^{- {\omega_{w}{({t_{k} - t_{k - 1}})}}}{\sum\limits_{l = 1}^{k - 1}{{p_{l}\left\lbrack {s_{M}^{\mu_{l} - 1}s_{N}^{v_{l} - 1^{T}}} \right\rbrack}e^{- {\omega_{w}{({t_{k - 1} - t_{l}})}}}}}}}} & (153) \\ {{{= {e^{- {\omega_{\omega}{({t_{k} - t_{k - 1}})}}}{\overset{\sim}{X}\left( {t_{k - 1} + 0} \right)}}};}} & (154) \end{matrix}$ upon arrival of event τ=t_(k-1),

$\begin{matrix} {{\overset{\sim}{X}\left( {t_{k} + 0} \right)} = {\sum\limits_{l = 1}^{k - 1}{{p_{l}\left\lbrack {s_{M}^{\mu_{l} - 1}s_{N}^{v_{l} - 1^{T}}} \right\rbrack}e^{- {\omega_{w}{({\tau - t_{l}})}}}}}} & (155) \\ {{= {{p_{k}\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack} + {\sum\limits_{l = 1}^{k - 1}{{p_{l}\left\lbrack {s_{M}^{\mu_{l} - 1}s_{N}^{v_{l} - 1^{T}}} \right\rbrack}e^{- {\omega_{w}{({t_{k -}t_{l}})}}}}}}}} & (156) \\ {{= {{p_{k}\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack} + {{\overset{\sim}{X}\left( {t_{k - 1} - 0} \right)}.}}}} & (157) \end{matrix}$

Combining Eqns. 154 and 157, the change of {tilde over (X)} from event t_(k-1) to event t_(k), may be given by: {tilde over (X)}(t _(k)+0)=p _(k) [s _(M) ^(μ) ^(k) ⁻¹ s _(N) ^(ν) ^(k) ⁻¹ ^(T) ]+e ^(−ω) ^(ω) ^((t) ^(k-) ^(t) ^(k-1) ⁾ {tilde over (X)}(t _(k-1)+0)   (158)

Therefore, the purely event-driven technique described in Table 5 performs the computation of short-time spatial DFT of polarized event trains under causal exponential window function.

Table 5 includes exemplary pseudo code for an event-driven technique, which performs the computation of short-time spatial Discrete FT of polarized event trains under causal exponential window function. Table 5 generalizes the short-time Fourier transform (STFT) for a single function of time to multiple functions of time for spatial STFT (e.g., to process signals across 320×480 pixels.)

TABLE 5 Pseudocode Comment  1 (t, t_(last)) ← (t₀, t₀) Current and last event times  2 t ← [t₁, ..., t_(K)] Event time vector  3 p ← [p₁, ... , p_(K)] Event polarity vector  4 μ ← [μ₁, ... , μ_(K)] Event coordinate vector  5 ν ← [ν₁, ... , ν_(K)] Event coordinate vector  6 X ← O_(M×N) State variable vectors of length M × N  7 for k from 1 to K do For each event  8 t ← t_(k) Current event time  9 μ ← μ_(k) Current event coordinate 10 ν ← ν_(k) Current event coordinate 11 X ← p_(k)[s_(M) ^(μ−1)s_(N) ^(ν−1) ^(T) ] + e^(−ω) ^(ω) ^((t−t) ^(last) ⁾X Update X 12 t_(last) ← t Current event becomes last event 13 end for Discrete-Space-Continuous-Time Short-Time Fourier Transform

Further generalizations may be employed to produce a purely event-driven technique for computation of the full, spatio-temporal, STFT in discrete space and continuous time.

Spatio-Temporal STFT

Given signals X(t)=[x_(μ,ν)](t) (μ=1, . . . , M; ν=1, . . . , N), the spatio-temporal is essentially the instantaneous spatial DFT of the STFTs of components of X(t), e.g., x_(μ,ν)(t). That is to say, {tilde over (X)}(τ,ω)=[S _(M) {tilde over (x)} _(μ,ν) S _(N) ^(T)](τ,ω), and  (159) {tilde over (x)}(τ,ω)=∫x _(μ,ν)(t)ω(t−τ)e ^(−1ωt) dt;  (160) hence,

$\begin{matrix} {{\overset{\sim}{X}\left( {\tau,\omega} \right)} = {{S_{M}\left\lbrack {\int{{X(t)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}} \right\rbrack}S_{N}^{T}}} & (161) \\ {\mspace{85mu}{= {\int{S_{M}{X(t)}S_{N}^{T}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}}} & (162) \end{matrix}$ Event-Driven Spatio-Temporal STFT of APM Sampled Signals

If, ∀μ, ν, we APM sample signal x_(μ,ν)(t) under kernel function h(t) and generate polarized event trains Ξ(t;{(t_(k),p_(k),μ_(k),ν_(k))})=[ξμ,ν](t), the original signals X(t) can be approximated by direct reconstructions

$\begin{matrix} {{Y(t)} = {{\sum\limits_{k}{p_{k}\Delta_{\mu_{k},v_{k}}{h\left( {t - t_{k}} \right)}}} \approx {X(t)}}} & (163) \end{matrix}$

Substituting X(t) with its approximation Y(t) in Eqn. 162, results in an approximation of {tilde over (X)}(τ,ω),

$\begin{matrix} {{\overset{\sim}{Y}\left( {\tau,\omega} \right)} = {\int{S_{M}{Y(t)}S_{N}^{T}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}} & (164) \\ {\mspace{79mu}{= {\int{S_{m}{\sum\limits_{k}{p_{k}\Delta_{\mu_{k},v_{k}}{h\left( {t - t_{k}} \right)}S_{N}^{T}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}}}}} & (165) \\ {\mspace{79mu}{= {\sum\limits_{k}{p_{k}S_{M}\Delta_{\mu_{k},v_{k}}S_{N}^{T}{\int{{h\left( {t - t_{k}} \right)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}}}}} & (166) \\ {\mspace{79mu}{= {\sum\limits_{k}{{p_{k}\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack}{\int{{h\left( {t - t_{k}} \right)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\;{t.}}}}}}} & (167) \end{matrix}$ Defining {tilde over (Y)} _(k)(τ,ω)=[s _(M) ^(μ) ^(k) ⁻¹ s _(N) ^(ν) ^(k) ⁻¹ ^(T) ]∫h(t−t _(k))ω(t−τ)e ^(−iωt) dt  (168) produces

$\begin{matrix} {{{\overset{\sim}{Y}}_{k}\left( {\tau,\omega} \right)} = {\sum\limits_{k}{p_{k}{{\overset{\sim}{Y}}_{k}\left( {\tau,\omega} \right)}}}} & (169) \end{matrix}$ Causal Exponential Encoding Pulse and STFT Window

Where both APM encoding kernel and STFT window are causal exponential functions for, i Eqns. 25 and 26, Eqn. 168 becomes

$\begin{matrix} {{{\overset{\sim}{Y}}_{k}\left( {\tau,\omega} \right)} = {\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack{\int{{h\left( {t - t_{k}} \right)}{\omega\left( {t - \tau} \right)}e^{{- i}\;\omega\; t}d\; t}}}} & (170) \\ \begin{matrix} {\mspace{85mu}{= {\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack{\int{{\Theta\left( {t - t_{k}} \right)}{\Theta\left( {\tau - t} \right)}}}}}} \\ {e^{{{- \omega_{\omega}}\tau} + {\omega_{h}t_{k}} - {{({\omega_{h} - \omega_{w} + {i\;\omega}})}t}}d\; t} \end{matrix} & (171) \\ \begin{matrix} {\mspace{85mu}{= {\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack{\Theta\left( {\tau - t_{k}} \right)}e^{{{- \omega_{w}}\tau} + {\omega_{h}t_{k}}}}}} \\ {\int_{t_{k}}^{T}{e^{{- {({\omega_{h} - \omega_{w} + {i\;\omega}})}}t}d\; t}} \end{matrix} & (172) \\ \begin{matrix} {\mspace{85mu}{= \frac{\left\lbrack {s_{M}^{\mu_{k} - 1}s_{N}^{v_{k} - 1^{T}}} \right\rbrack{\Theta\left( {\tau - t_{k}} \right)}}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}}}} \\ {\left\{ {e^{{- {\omega_{h}{({\tau - t_{k}})}}} - {i{\lbrack{{\omega\; t} + {\tau{(\omega)}} - \pi}\rbrack}}} + e^{{- {\omega_{w}{({\tau - t_{k}})}}} - {i{\lbrack{{\omega\; t_{k}} + {y{(\omega)}}}\rbrack}}}} \right\}} \end{matrix} & (173) \end{matrix}$ An Event-Driven Technique for Full Spatio-Temporal STFT with Causal Exponential APM Encoding Pulse and STFT Window

Similar to derivations presented above, we arrive at a purely event-driven technique for full spatio-temporal STFT in discrete space and continuous time, for exponential APM sampled signals under an exponential STFT window. The technique is summarized in Table 6.

Table 6 includes exemplary pseudo code for an event-driven technique for full spatio-temporal STFT in discrete space and continuous time, for exponential APM sampled signals under an exponential STFT window.

TABLE 6 Pseudocode Comment 1 (t, t_(last)) ← (t_(o), t_(o)) Current and last event times 2 t ← [t₁, . . . , t_(K)] Event time vector 3 p ← [p₁, . . . , p_(K)] Event polarity vector 4 μ ← [μ₁, . . . , μ_(K)] Event coordinate vector 5 v ← [v₁, . . . , v_(K)] Event coordinate vector 6 ω ← [ω₁, . . . , ω_(L)] Frequency vector 7 $\left. \gamma\leftarrow{\tan^{- 1}\frac{\omega}{\omega_{h} - \omega_{w}}} \right.$ Phase offset vector 8 (U,V,Y) ← (O_(L×M×N), O_(L×M×N), O_(L×M×N)) State variable matrices of size L × M × N 9 for k from 1 to K do For each event 10  t ← t_(k) Current event time 11  μ ← μ_(k) Current event coordinate 12  v ← v_(k) Current event coordinate 13  for l from 1 to K do For each frequency 14   (ω, γ) ← (ω_(l), γ_(l)) Current frequency 15   $\left. U_{l}\leftarrow{\frac{{p_{k}\left\lbrack {s_{M}^{\mu - 1}s_{N}^{v - 1^{T}}} \right\rbrack}e^{- {i{({{\omega\; t} + \gamma - \pi})}}}}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}} + {e^{{- {({\omega_{h} + {i\;\omega}})}}{({t - t_{last}})}}U_{l}}} \right.$ Update U 16   $\left. V_{l}\leftarrow{\frac{{p_{k}\left\lbrack {s_{M}^{\mu - 1}s_{N}^{v - 1^{T}}} \right\rbrack}e^{- {i{({{\omega\; t} + \gamma})}}}}{\sqrt{\left( {\omega_{h} - \omega_{w}} \right)^{2} + \omega^{2}}} + {e^{- {\omega_{w}{({t - t_{last}})}}}V_{l}}} \right.$ Update V 17  end for 18  Y ← U + V Update Y 19  t_(last) ← t Current event becomes last event 20 end for

FIG. 1 illustrates an example implementation of the aforementioned processing asynchronous event driven samples of a continuous time signal using a system-on-a-chip (SOC) 100, which may include a general-purpose processor (CPU) or multi-core general-purpose processors (CPUs) 102 in accordance with certain aspects of the present disclosure. Variables (e.g., neural signals and synaptic weights), system parameters associated with a computational device (e.g., neural network with weights), delays, frequency bin information, and task information may be stored in a memory block associated with a neural processing unit (NPU) 108, in a memory block associated with a CPU 102, in a memory block associated with a graphics processing unit (GPU) 104, in a memory block associated with a digital signal processor (DSP) 106, in a dedicated memory block 118, or may be distributed across multiple blocks. Instructions executed at the general-purpose processor 102 may be loaded from a program memory associated with the CPU 102 or may be loaded from a dedicated memory block 118.

The SOC 100 may also include additional processing blocks tailored to specific functions, such as a GPU 104, a DSP 106, a connectivity block 110, which may include fourth generation long term evolution (4G LTE) connectivity, unlicensed Wi-Fi connectivity, USB connectivity, Bluetooth connectivity, and the like, and a multimedia processor 112 such as an image capturing device that may, for example, detect and recognize gestures. In one implementation, the NPU is implemented in the CPU, DSP, and/or GPU. The SOC 100 may also include a sensor processor 114, image signal processors (ISPs), and/or navigation 120, which may include a global positioning system. The SOC may be based on an ARM instruction set. In an aspect of the present disclosure, the instructions loaded into the general-purpose processor 102 may comprise code for calculating a short-time Fourier transform output based on event driven samples.

FIG. 2 illustrates an example implementation of a system 200 in accordance with certain aspects of the present disclosure. As illustrated in FIG. 2, the system 200 may have multiple local processing units 202 that may perform various operations of methods described herein. Each local processing unit 202 may comprise a local state memory 204 and a local parameter memory 206 that may store parameters of a neural network. In addition, the local processing unit 202 may have a local (neuron) model program (LMP) memory 208 for storing a local model program, a local learning program (LLP) memory 210 for storing a local learning program, and a local connection memory 212. Furthermore, as illustrated in FIG. 2, each local processing unit 202 may interface with a configuration processor unit 214 for providing configurations for local memories of the local processing unit, and with a routing connection processing unit 216 that provides routing between the local processing units 202.

Deep learning architectures may perform an object recognition task by learning to represent inputs at successively higher levels of abstraction in each layer, thereby building up a useful feature representation of the input data. In this way, deep learning addresses a major bottleneck of traditional machine learning. Prior to the advent of deep learning, a machine learning approach to an object recognition problem may have relied heavily on human engineered features, perhaps in combination with a shallow classifier. A shallow classifier may be a two-class linear classifier, for example, in which a weighted sum of the feature vector components may be compared with a threshold to predict to which class the input belongs. Human engineered features may be templates or kernels tailored to a specific problem domain by engineers with domain expertise. Deep learning architectures, in contrast, may learn to represent features that are similar to what a human engineer might design, but through training. Furthermore, a deep network may learn to represent and recognize new types of features that a human might not have considered.

A deep learning architecture may learn a hierarchy of features. If presented with visual data, for example, the first layer may learn to recognize simple features, such as edges, in the input stream. If presented with auditory data, the first layer may learn to recognize spectral power in specific frequencies. The second layer, taking the output of the first layer as input, may learn to recognize combinations of features, such as simple shapes for visual data or combinations of sounds for auditory data. Higher layers may learn to represent complex shapes in visual data or words in auditory data. Still higher layers may learn to recognize common visual objects or spoken phrases.

Deep learning architectures may perform especially well when applied to problems that have a natural hierarchical structure. For example, the classification of motorized vehicles may benefit from first learning to recognize wheels, windshields, and other features. These features may be combined at higher layers in different ways to recognize cars, trucks, and airplanes.

Neural networks may be designed with a variety of connectivity patterns. In feed-forward networks, information is passed from lower to higher layers, with each neuron in a given layer communicating to neurons in higher layers. A hierarchical representation may be built up in successive layers of a feed-forward network, as described above. Neural networks may also have recurrent or feedback (also called top-down) connections. In a recurrent connection, the output from a neuron in a given layer is communicated to another neuron in the same layer. A recurrent architecture may be helpful in recognizing patterns that unfold in time. A connection from a neuron in a given layer to a neuron in a lower layer is called a feedback (or top-down) connection. A network with many feedback connections may be helpful when the recognition of a high level concept may aid in discriminating the particular low-level features of an input.

Referring to FIG. 3A, the connections between layers of a neural network may be fully connected 302 or locally connected 304. In a fully connected network 302, a neuron in a given layer may communicate its output to every neuron in the next layer. Alternatively, in a locally connected network 304, a neuron in a given layer may be connected to a limited number of neurons in the next layer. A convolutional network 306 may be locally connected, and is furthermore a special case in which the connection strengths associated with each neuron in a given layer are shared (e.g., 308). More generally, a locally connected layer of a network may be configured so that each neuron in a layer will have the same or a similar connectivity pattern, but with connections strengths that may have different values (e.g., 310, 312, 314, and 316). The locally connected connectivity pattern may give rise to spatially distinct receptive fields in a higher layer, because the higher layer neurons in a given region may receive inputs that are tuned through training to the properties of a restricted portion of the total input to the network.

Locally connected neural networks may be well suited to problems in which the spatial location of inputs is meaningful. For instance, a network 300 designed to recognize visual features from a car-mounted camera may develop high layer neurons with different properties depending on their association with the lower versus the upper portion of the image. Neurons associated with the lower portion of the image may learn to recognize lane markings, for example, while neurons associated with the upper portion of the image may learn to recognize traffic lights, traffic signs, and the like.

A DCN may be trained with supervised learning. During training, a DCN may be presented with an image 326, such as a cropped image of a speed limit sign as shown, and a “forward pass” may then be computed to produce an output 328. The output 328 may be a vector of values corresponding to features such as “sign,” “60,” and “100.” The network designer may want the DCN to output a high score for some of the neurons in the output feature vector, for example the ones corresponding to “sign” and “60” as shown in the output 328 for a network 300 that has been trained. Before training, the output produced by the DCN is likely to be incorrect, and so an error may be calculated between the actual output and the target output. The weights of the DCN may then be adjusted so that the output scores of the DCN are more closely aligned with the target.

To properly adjust the weights, a learning process may compute a gradient vector for the weights. The gradient may indicate an amount that an error would increase or decrease if the weight were adjusted slightly. At the top layer, the gradient may correspond directly to the value of a weight connecting an activated neuron in the penultimate layer and a neuron in the output layer. In lower layers, the gradient may depend on the value of the weights and on the computed error gradients of the higher layers. The weights may then be adjusted so as to reduce the error. This manner of adjusting the weights may be referred to as “back propagation” as it involves a “backward pass” through the neural network.

In practice, the error gradient of weights may be calculated over a small number of examples, so that the calculated gradient approximates the true error gradient. This approximation method may be referred to as stochastic gradient descent. Stochastic gradient descent may be repeated until the achievable error rate of the entire system has stopped decreasing or until the error rate has reached a target level.

After learning, the DCN may be presented with new images 326 and a forward pass through the network may yield an output 328 that may be considered an inference or a prediction of the DCN.

Deep belief networks (DBNs) are probabilistic models comprising multiple layers of hidden nodes. DBNs may be used to extract a hierarchical representation of training data sets. A DBN may be obtained by stacking up layers of Restricted Boltzmann Machines (RBMs). An RBM is a type of artificial neural network that can learn a probability distribution over a set of inputs. Because RBMs can learn a probability distribution in the absence of information about the class to which each input should be categorized, RBMs are often used in unsupervised learning. Using a hybrid unsupervised and supervised paradigm, the bottom RBMs of a DBN may be trained in an unsupervised manner and may serve as feature extractors, and the top RBM may be trained in a supervised manner (on a joint distribution of inputs from the previous layer and target classes) and may serve as a classifier.

Deep convolutional networks (DCNs) are networks of convolutional networks, configured with additional pooling and normalization layers. DCNs have achieved state-of-the-art performance on many tasks. DCNs can be trained using supervised learning in which both the input and output targets are known for many exemplars and are used to modify the weights of the network by use of gradient descent methods.

DCNs may be feed-forward networks. In addition, as described above, the connections from a neuron in a first layer of a DCN to a group of neurons in the next higher layer are shared across the neurons in the first layer. The feed-forward and shared connections of DCNs may be exploited for fast processing. The computational burden of a DCN may be much less, for example, than that of a similarly sized neural network that comprises recurrent or feedback connections.

The processing of each layer of a convolutional network may be considered a spatially invariant template or basis projection. If the input is first decomposed into multiple channels, such as the red, green, and blue channels of a color image, then the convolutional network trained on that input may be considered three-dimensional, with two spatial dimensions along the axes of the image and a third dimension capturing color information. The outputs of the convolutional connections may be considered to form a feature map in the subsequent layer 318, 320, and 322, with each element of the feature map (e.g., 320) receiving input from a range of neurons in the previous layer (e.g., 318) and from each of the multiple channels. The values in the feature map may be further processed with a non-linearity, such as a rectification, max(0,x). Values from adjacent neurons may be further pooled 324, which corresponds to down sampling, and may provide additional local invariance and dimensionality reduction. Normalization, which corresponds to whitening, may also be applied through lateral inhibition between neurons in the feature map.

The performance of deep learning architectures may increase as more labeled data points become available or as computational power increases. Modern deep neural networks are routinely trained with computing resources that are thousands of times greater than what was available to a typical researcher just fifteen years ago. New architectures and training paradigms may further boost the performance of deep learning. Rectified linear units may reduce a training issue known as vanishing gradients. New training techniques may reduce over-fitting and thus enable larger models to achieve better generalization. Encapsulation techniques may abstract data in a given receptive field and further boost overall performance.

FIG. 3B is a block diagram illustrating an exemplary deep convolutional network 350. The deep convolutional network 350 may include multiple different types of layers based on connectivity and weight sharing. As shown in FIG. 3B, the exemplary deep convolutional network 350 includes multiple convolution blocks (e.g., C1 and C2). Each of the convolution blocks may be configured with a convolution layer, a normalization layer (LNorm), and a pooling layer. The convolution layers may include one or more convolutional filters, which may be applied to the input data to generate a feature map. Although only two convolution blocks are shown, the present disclosure is not so limiting, and instead, any number of convolutional blocks may be included in the deep convolutional network 350 according to design preference. The normalization layer may be used to normalize the output of the convolution filters. For example, the normalization layer may provide whitening or lateral inhibition. The pooling layer may provide down sampling aggregation over space for local invariance and dimensionality reduction.

The parallel filter banks, for example, of a deep convolutional network may be loaded on a CPU 102 or GPU 104 of an SOC 100, optionally based on an ARM instruction set, to achieve high performance and low power consumption. In alternative embodiments, the parallel filter banks may be loaded on the DSP 106 or an ISP 116 of an SOC 100. In addition, the DCN may access other processing blocks that may be present on the SOC, such as processing blocks dedicated to sensors 114 and navigation 120.

The deep convolutional network 350 may also include one or more fully connected layers (e.g., FC1 and FC2). The deep convolutional network 350 may further include a logistic regression (LR) layer. Between each layer of the deep convolutional network 350 are weights (not shown) that are to be updated. The output of each layer may serve as an input of a succeeding layer in the deep convolutional network 350 to learn hierarchical feature representations from input data (e.g., images, audio, video, sensor data and/or other input data) supplied at the first convolution block C1.

FIG. 4 is a block diagram illustrating an exemplary software architecture 400 that may modularize artificial intelligence (AI) functions. Using the architecture, applications 402 may be designed that may cause various processing blocks of an SOC 420 (for example a CPU 422, a DSP 424, a GPU 426 and/or an NPU 428) to perform supporting computations during run-time operation of the application 402.

The AI application 402 may be configured to call functions defined in a user space 404 that may, for example, provide for the detection and recognition of a scene indicative of the location in which the device currently operates. The AI application 402 may, for example, configure a microphone and a camera differently depending on whether the recognized scene is an office, a lecture hall, a restaurant, or an outdoor setting such as a lake. The AI application 402 may make a request to compiled program code associated with a library defined in a SceneDetect application programming interface (API) 406 to provide an estimate of the current scene. This request may ultimately rely on the output of a deep neural network configured to provide scene estimates based on video and positioning data, for example.

A run-time engine 408, which may be compiled code of a Runtime Framework, may be further accessible to the AI application 402. The AI application 402 may cause the run-time engine, for example, to request a scene estimate at a particular time interval or triggered by an event detected by the user interface of the application. When caused to estimate the scene, the run-time engine may in turn send a signal to an operating system 410, such as a Linux Kernel 412, running on the SOC 420. The operating system 410, in turn, may cause a computation to be performed on the CPU 422, the DSP 424, the GPU 426, the NPU 428, or some combination thereof. The CPU 422 may be accessed directly by the operating system, and other processing blocks may be accessed through a driver, such as a driver 414-418 for a DSP 424, for a GPU 426, or for an NPU 428. In the exemplary example, the deep neural network may be configured to run on a combination of processing blocks, such as a CPU 422 and a GPU 426, or may be run on an NPU 428, if present.

FIG. 5 is a block diagram illustrating the run-time operation 500 of an AI application on a smartphone 502. The AI application may include a pre-process module 504 that may be configured (using for example, the JAVA programming language) to convert the format of an image 506 and then crop and/or resize the image 508. The pre-processed image may then be communicated to a classify application 510 that contains a SceneDetect Backend Engine 512 that may be configured (using for example, the C programming language) to detect and classify scenes based on visual input. The SceneDetect Backend Engine 512 may be configured to further preprocess 514 the image by scaling 516 and cropping 518. For example, the image may be scaled and cropped so that the resulting image is 224 pixels by 224 pixels. These dimensions may map to the input dimensions of a neural network. The neural network may be configured by a deep neural network block 520 to cause various processing blocks of the SOC 100 to further process the image pixels with a deep neural network. The results of the deep neural network may then be thresholded 522 and passed through an exponential smoothing block 524 in the classify application 510. The smoothed results may then cause a change of the settings and/or the display of the smartphone 502.

In one configuration, a neuron model is configured for calculating a short-time Fourier transform output based on event driven samples and/or interpolating output between events. The neuron model includes a calculating means and/or interpolating means. In one aspect, the calculating means and/or interpolating means may be the general-purpose processor 102, program memory associated with the general-purpose processor 102, memory block 118, local processing units 202, and or the routing connection processing units 216 configured to perform the functions recited. In another configuration, the aforementioned means may be any module or any apparatus configured to perform the functions recited by the aforementioned means.

According to certain aspects of the present disclosure, each local processing unit 202 may be configured to determine parameters of the neural network based upon desired one or more functional features of the neural network, and develop the one or more functional features towards the desired functional features as the determined parameters are further adapted, tuned and updated.

FIG. 6 illustrates a method 600 for processing asynchronous event driven samples of a continuous time signal. In block 602, the process calculates a short-time Fourier transform (STFT) output based on event driven samples. In some aspects, the process calculates the STFT output by expressing an encoding pulse and an STFT window function as a sum of complex weighted causal complex exponentials.

In other aspects, the STFT output may be calculated by approximating an encoding pulse in an STFT window function as a sum of complex weighted causal complex exponentials. The encoding pulse may comprise an asynchronous pulse modulated (APM) encoding pulse. Furthermore, in block 604, the process interpolates output between events.

In some aspects, the process may further include calculating a spatio-temporal STFT output. The spatio-temporal STFT output may be calculated based on an event driven spatial discrete Fourier transform (DFT) output using continuous time event driven STFT outputs.

FIG. 7 is a block diagram illustrating a method for event-driven short-time Fourier transform processing according to aspects of the present disclosure. In block 702, the process initializes state variables (e.g., u, ν). In block 704, the process represents a continuous time input signal as a sequence of events. For example, an event-based sampling process (e.g., APM sampling) may be applied to generate a train of pulses. In some aspects, the train of pulses may be positive pulses, negative pulses or both (bipolar).

In block 706, the process determines whether an event has occurred. If an event has occurred, the process updates the state variables (e.g., u, ν) for each frequency in the time window in block 708. In block 710, the process calculates the output signal (e.g., STFT output, spatio-temporal STFT or DFT output) at time t_(k) using the updated state variables. The output signal may comprise an approximation of the continuous time output signal y(t). On the other hand, if an event has not occurred, the process may return to block 706 to await the occurrence of an event (e.g., remain in an idle mode).

The various operations of methods described above may be performed by any suitable means capable of performing the corresponding functions. The means may include various hardware and/or software component(s) and/or module(s), including, but not limited to, a circuit, an application specific integrated circuit (ASIC), or processor. Generally, where there are operations illustrated in the figures, those operations may have corresponding counterpart means-plus-function components with similar numbering.

As used herein, the term “determining” encompasses a wide variety of actions. For example, “determining” may include calculating, computing, processing, deriving, investigating, looking up (e.g., looking up in a table, a database or another data structure), ascertaining and the like. Additionally, “determining” may include receiving (e.g., receiving information), accessing (e.g., accessing data in a memory) and the like. Furthermore, “determining” may include resolving, selecting, choosing, establishing and the like.

As used herein, a phrase referring to “at least one of” a list of items refers to any combination of those items, including single members. As an example, “at least one of: a, b, or c” is intended to cover: a, b, c, a-b, a-c, b-c, and a-b-c.

The various illustrative logical blocks, modules and circuits described in connection with the present disclosure may be implemented or performed with a general-purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array signal (FPGA) or other programmable logic device (PLD), discrete gate or transistor logic, discrete hardware components or any combination thereof designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but in the alternative, the processor may be any commercially available processor, controller, microcontroller or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.

The steps of a method or algorithm described in connection with the present disclosure may be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. A software module may reside in any form of storage medium that is known in the art. Some examples of storage media that may be used include random access memory (RAM), read only memory (ROM), flash memory, erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, a hard disk, a removable disk, a CD-ROM and so forth. A software module may comprise a single instruction, or many instructions, and may be distributed over several different code segments, among different programs, and across multiple storage media. A storage medium may be coupled to a processor such that the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor.

The methods disclosed herein comprise one or more steps or actions for achieving the described method. The method steps and/or actions may be interchanged with one another without departing from the scope of the claims. In other words, unless a specific order of steps or actions is specified, the order and/or use of specific steps and/or actions may be modified without departing from the scope of the claims.

The functions described may be implemented in hardware, software, firmware, or any combination thereof. If implemented in hardware, an example hardware configuration may comprise a processing system in a device. The processing system may be implemented with a bus architecture. The bus may include any number of interconnecting buses and bridges depending on the specific application of the processing system and the overall design constraints. The bus may link together various circuits including a processor, machine-readable media, and a bus interface. The bus interface may be used to connect a network adapter, among other things, to the processing system via the bus. The network adapter may be used to implement signal processing functions. For certain aspects, a user interface (e.g., keypad, display, mouse, joystick, etc.) may also be connected to the bus. The bus may also link various other circuits such as timing sources, peripherals, voltage regulators, power management circuits, and the like, which are well known in the art, and therefore, will not be described any further.

The processor may be responsible for managing the bus and general processing, including the execution of software stored on the machine-readable media. The processor may be implemented with one or more general-purpose and/or special-purpose processors. Examples include microprocessors, microcontrollers, DSP processors, and other circuitry that can execute software. Software shall be construed broadly to mean instructions, data, or any combination thereof, whether referred to as software, firmware, middleware, microcode, hardware description language, or otherwise. Machine-readable media may include, by way of example, random access memory (RAM), flash memory, read only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable Read-only memory (EEPROM), registers, magnetic disks, optical disks, hard drives, or any other suitable storage medium, or any combination thereof. The machine-readable media may be embodied in a computer-program product. The computer-program product may comprise packaging materials.

In a hardware implementation, the machine-readable media may be part of the processing system separate from the processor. However, as those skilled in the art will readily appreciate, the machine-readable media, or any portion thereof, may be external to the processing system. By way of example, the machine-readable media may include a transmission line, a carrier wave modulated by data, and/or a computer product separate from the device, all which may be accessed by the processor through the bus interface. Alternatively, or in addition, the machine-readable media, or any portion thereof, may be integrated into the processor, such as the case may be with cache and/or general register files. Although the various components discussed may be described as having a specific location, such as a local component, they may also be configured in various ways, such as certain components being configured as part of a distributed computing system.

The processing system may be configured as a general-purpose processing system with one or more microprocessors providing the processor functionality and external memory providing at least a portion of the machine-readable media, all linked together with other supporting circuitry through an external bus architecture. Alternatively, the processing system may comprise one or more neuromorphic processors for implementing the neuron models and models of neural systems described herein. As another alternative, the processing system may be implemented with an application specific integrated circuit (ASIC) with the processor, the bus interface, the user interface, supporting circuitry, and at least a portion of the machine-readable media integrated into a single chip, or with one or more field programmable gate arrays (FPGAs), programmable logic devices (PLDs), controllers, state machines, gated logic, discrete hardware components, or any other suitable circuitry, or any combination of circuits that can perform the various functionality described throughout this disclosure. Those skilled in the art will recognize how best to implement the described functionality for the processing system depending on the particular application and the overall design constraints imposed on the overall system.

The machine-readable media may comprise a number of software modules. The software modules include instructions that, when executed by the processor, cause the processing system to perform various functions. The software modules may include a transmission module and a receiving module. Each software module may reside in a single storage device or be distributed across multiple storage devices. By way of example, a software module may be loaded into RAM from a hard drive when a triggering event occurs. During execution of the software module, the processor may load some of the instructions into cache to increase access speed. One or more cache lines may then be loaded into a general register file for execution by the processor. When referring to the functionality of a software module below, it will be understood that such functionality is implemented by the processor when executing instructions from that software module. Furthermore, it should be appreciated that aspects of the present disclosure result in improvements to the functioning of the processor, computer, machine, or other system implementing such aspects.

If implemented in software, the functions may be stored or transmitted over as one or more instructions or code on a computer-readable medium. Computer-readable media include both computer storage media and communication media including any medium that facilitates transfer of a computer program from one place to another. A storage medium may be any available medium that can be accessed by a computer. By way of example, and not limitation, such computer-readable media can comprise RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to carry or store desired program code in the form of instructions or data structures and that can be accessed by a computer. Additionally, any connection is properly termed a computer-readable medium. For example, if the software is transmitted from a website, server, or other remote source using a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared (IR), radio, and microwave, then the coaxial cable, fiber optic cable, twisted pair, DSL, or wireless technologies such as infrared, radio, and microwave are included in the definition of medium. Disk and disc, as used herein, include compact disc (CD), laser disc, optical disc, digital versatile disc (DVD), floppy disk, and Blu-ray® disc where disks usually reproduce data magnetically, while discs reproduce data optically with lasers. Thus, in some aspects computer-readable media may comprise non-transitory computer-readable media (e.g., tangible media). In addition, for other aspects computer-readable media may comprise transitory computer-readable media (e.g., a signal). Combinations of the above should also be included within the scope of computer-readable media.

Thus, certain aspects may comprise a computer program product for performing the operations presented herein. For example, such a computer program product may comprise a computer-readable medium having instructions stored (and/or encoded) thereon, the instructions being executable by one or more processors to perform the operations described herein. For certain aspects, the computer program product may include packaging material.

Further, it should be appreciated that modules and/or other appropriate means for performing the methods and techniques described herein can be downloaded and/or otherwise obtained by a user terminal and/or base station as applicable. For example, such a device can be coupled to a server to facilitate the transfer of means for performing the methods described herein. Alternatively, various methods described herein can be provided via storage means (e.g., RAM, ROM, a physical storage medium such as a compact disc (CD) or floppy disk, etc.), such that a user terminal and/or base station can obtain the various methods upon coupling or providing the storage means to the device. Moreover, any other suitable technique for providing the methods and techniques described herein to a device can be utilized.

It is to be understood that the claims are not limited to the precise configuration and components illustrated above. Various modifications, changes and variations may be made in the arrangement, operation and details of the methods and apparatus described above without departing from the scope of the claims. 

What is claimed is:
 1. A method of processing a plurality of asynchronous event driven samples of a continuous analog time signal in a computational network, comprising: receiving the continuous analog time signal at a vision sensor; generating the plurality of asynchronous event driven samples based on event-driven sampling of the continuous analog time signal; calculating a short-time Fourier transform (STFT) digital signal directly from the plurality of asynchronous event driven samples of the continuous analog time signal; and processing the STFT digital signal.
 2. The method of claim 1, further comprising interpolating STFT digital signals between events.
 3. The method of claim 1, in which the calculating comprises expressing an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 4. The method of claim 1, in which the calculating comprises approximating an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 5. The method of claim 4, in which the asynchronous event driven sample comprises an asynchronous pulse modulated (APM) encoding sample.
 6. The method of claim 1, further comprising calculating a spatio-temporal STFT digital signal based at least in part on an event driven spatial discrete Fourier transform (DFT) digital signal using a plurality of continuous time event driven STFT digital signals.
 7. An image capturing device, the image capturing device comprising: a memory unit; and at least one processor coupled to the memory unit, the at least one processor configured: to receive a continuous analog time signal at a vision sensor; to generate a plurality of asynchronous event driven samples based on event-driven sampling of the continuous analog time signal; to calculate a short-time Fourier transform (STFT) digital signal directly from the plurality of asynchronous event driven samples of the continuous analog time signal, the event driven samples comprising images; and to output the STFT digital signal.
 8. The image capturing device of claim 7, in which the at least one processor is further configured to interpolate the STFT digital signal between events.
 9. The image capturing device of claim 7, in which the at least one processor is further configured to express an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 10. The image capturing device of claim 7, in which the at least one processor is further configured to approximate an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 11. The image capturing device of claim 10, in which the asynchronous event driven sample comprises an asynchronous pulse modulated (APM) sample.
 12. The image capturing device of claim 7, in which the at least one processor is further configured to calculate a spatio-temporal STFT digital signal based at least in part on an event driven spatial discrete Fourier transform (DFT) digital signal using a plurality of continuous time event driven STFT digital signals.
 13. An apparatus for processing a plurality of asynchronous event driven samples of a continuous analog time signal, comprising: means for receiving the continuous analog time signal; means for generating the plurality of asynchronous event driven samples based on event-driven sampling of the continuous analog time signal; means for calculating a short-time Fourier transform (STFT) digital signal directly from the plurality of asynchronous event driven samples of the continuous analog time signal; and means for processing the STFT digital signal.
 14. The apparatus of claim 13, in which the means for calculating calculates the STFT digital signal by expressing an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 15. The apparatus of claim 13, in which the means for calculating calculates the STFT digital signal by approximating an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 16. The apparatus of claim 15, in which the asynchronous event driven sample comprises an asynchronous pulse modulated (APM) sample.
 17. The apparatus of claim 13, further comprising means for calculating a spatio-temporal STFT digital signal based at least in part on an event driven spatial discrete Fourier transform (DFT) digital signal using a plurality of continuous time event driven STFT digital signals.
 18. A non-transitory computer readable medium having program code recorded thereon for processing a plurality of asynchronous event driven samples of a continuous analog time signal, the program code executed by a processor and comprising: program code to receive the continuous analog time signal at a vision sensor; program code to generate the plurality of asynchronous event driven samples based on event-driven sampling of the continuous analog time signal; program code to calculate a short-time Fourier transform (STFT) digital signal directly from the plurality of asynchronous event driven samples of the continuous analog time signal; and program code to process the STFT digital signal.
 19. The non-transitory computer readable medium of claim 18, in which the program code further comprises program code to interpolate the STFT digital signal between events.
 20. The non-transitory computer readable medium of claim 18, in which the program code to calculate further comprises program code to express an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 21. The non-transitory computer readable medium of claim 18, in which the program code to calculate further comprises program code to approximate an asynchronous event driven sample of the plurality of asynchronous event driven samples and an STFT window function as a sum of complex weighted causal complex exponentials.
 22. The non-transitory computer readable medium of claim 21, in which the asynchronous event driven sample comprises an asynchronous pulse modulated (APM) sample.
 23. The non-transitory computer readable medium of claim 18, in which the program code further comprises program code to calculate a spatio-temporal STFT digital signal based at least in part on an event driven spatial discrete Fourier transform (DFT) digital signal using a plurality of continuous time event driven STFT digital signals. 